Exercises 1.2: Definition of Polynomials b a b a b a b x …€¦Unit 1 Polynomials Exercises MHF 4UI Page 1 Exercises 1.1: Algebra Review 1. Simplify. ... poly-nomial if the x5 term

Unit 1 Polynomials Exercises MHF 4UI Page 1

Exercises 1.1: Algebra Review

1. Simplify. State any necessary restrictions on thefinal expression for it to be valid.

(a) 3(x+ 2)2 − 5(x− 3)2

(b) 2(b− a)(a+ 2b)− a(a− 3b) + 4b(2a+ 3b)

(c)(x(x− 2)2 − 1

)2+ x

(x2 + 3

)2(d)

x+ y

3− 2x− 3y

8

(e)x+ y

x− y+

x− y2(x+ y)

(f)a+ b

(a− b)2+

(a− b)2

a+ b

(g)1

a+ b+

1

a− b

2. Simplify. State any necessary restrictions on thefinal expression for it to be valid.

(a)a+ 1

a− a− 1

a+ 1+

a

a− 1

(b)ab −

ba

ba + a

b

+ab + b

aba −

ab

(c)a+bb−a −

b−aa+b

b−aa+b + a+b

b−a

+a+bb−a + b−a

a+bb+aa−b −

a−bb+a

3. Substitute x = a+ b and y = a− b into each of thefollowing and simplify.

(a)3x2(y − x)2

(x+ y)2

(b) (x+ y)(x−y)

(c) (x+ y)x+yx−y

Exercises 1.2: Definition of Polynomials

1. Which of the following are polynomial expressions?For each polynomial, state the degree and state ifthe polynomial is constant, linear, quadratic, cubicor quartic.

(a) 3x2 + 5

(b) −7

2x4 − x3 + 11x2 + x− 9

(c) 4x

(d) 5xn

(e)1

x3

(f) sin 3x

(g) πx2 + 4x− 7

(h) xn + xn−1 + xn−2 + · · ·+ x+ 1

(i) tanπx

4

(j) 6

(k)x+ 3

x− 2

(l) x2 + x+ 3− 1

x− 1

x2

2. Determine the domain and range of each of the fol-lowing polynomial functions. Also determine thebehaviour of each function as x approaches positiveand negative infinity.

(a) y = x+ 3

(b) y = 3x2 − 7

(c) y = −5x3

(d) y = x2 + 5x− 6


(e) y = (x+ 1)(x− 2)(x− 5)

(f) y = x3 − 2x2 − 7x+ 1

(g) y = −15x5 + 12x4 + 9x3 − 8x2 + x− 14

(h) y = x3 + x2 − x− 1

3. Draw all possible shapes for each of the followingpossible polynomial functions.

(a) Linear.

(b) Quadratic.

(c) Cubic.

(d) Quartic.

4. Determine the behaviour for very large positive or

negative x−values for a quintic (5th degree) poly-nomial if the x5 term is:

(a) Positive.

(b) Negative.

5. Explain why y = sinx is not a polynomial.

6. How can you quickly determine without doing anyalgebra that y = −(x + 1)(x − 3)(x + 2) and y =−x3 + 4x2 + x− 6 are two different polynomials?

7. (*) Application of polynomials: Multiply the num-bers 178932158473627 by 6487719321425 by mul-tiplying the related polynomials 178x4 + 932x3 +158x2+473x+627 and 6x4+487x3+719x2+321x+425 and then evaluating the product at x = 103.

8. (**) Find all periodic polynomials. That is, find allpolynomials y = f(x) such that f(x+k) = f(x) forsome constant k and all x ∈ R. From this result,show that y = sinx, y = cosx and y = tanx are notpolynomials. Hint: examine the difference functionf(x + k) − f(x) and recall that (a + b)n = an +

nan−1b+ n(n−1)2 an−2b2+ n(n−1)(n−2)

(2)(3) an−3b3+· · ·+bn,then simplify the result.

Exercises 1.3: Finite Differences for Polyno-mials

1. Using finite differences, determine the equation ofthe polynomial function given the data points pro-vided.

(a) (1,−3), (2,−7), (3,−3), (4, 15), (5, 53), (6, 117)

(b) (1,−1), (2,−4), (3,−15), (4,−38), (5,−77)

(c) (0, 0), (1, 1), (2, 4), (3, 13)

(d) (−2,−37), (−1, 2)(0, 5), (1, 2), (2,−49)

(e) (−1, 6), (0, 4), (1, 8), (2, 30)

(f) (0, 3), (1, 2), (2,−15), (3,−48), (4,−97)

(g) (1, 14), (2,−9), (3,−276), (4,−1165), (5,−3222)

2. (*) Define the operators:

∆f(x) = f(x+ 1)− f(x) (Difference)

If(x) = f(x) (Identity)

Ef(x) = f(x+ 1) (Shift)

(a) Determine an expression for ∆2f(x) =∆ (∆f(x)).

(b) Using ∆, ∆2 and ∆3, show that first, secondand third finite differences of linear, quadraticand cubic polynomials, respectively, are con-stant.

(c) Show that ∆ = E − I.

(d) Show that Ekf(x) = f(x+k) and that EpEq =Ep+q.

(e) Verify that Ek∆f(x) = ∆Ekf(x).

(f) Verify that the kth difference of an n-degreepolynomial is a polynomial of degree n− k.

3. (**) The operators defined in Exercise #2 maysometimes be used to find a formula for a finiteseries. Consider the series S =

∑n−1j=0 f(a + j) =


f(a) + f(a + 1) + f(a + 2) + · · · + f(a + (n − 1)).Rewriting S using difference operators and the iden-tity 1 +E +E2 + · · ·+En−1 = (En− I)/(E − I) =((1 + ∆)n − 1)/∆, we find that

S =

n−1∑j=0

Ejf(a)

=(1 + ∆)n − 1

∆f(a)

=

(n+

n(n− 1)

2∆ +

n(n− 1)(n− 2)

(2)(3)∆2 + · · ·

+∆n−1

)f(a).

(a) Use f(x) = x2 and a = 1 to construct a dif-ference table to complete the formula for S toshow that 1 + 4 + 9 + · · ·+n2 = n(n+ 1)(2n+1)/6.

(b) Find an explicit formula for 1+8+27+· · ·+n3.(c) Find an explicit formula for 3 + 8 + 15 + · · ·+

n(n+ 2).

Exercises 1.4: Common Factors and Factor-ing by Grouping

1. Factor fully.

(a) 6x3 + 9x

(b) 5x+ 25

(c) 11xy + 4x

(d) 75r − 25r

(e) 44mn+ 2m3

(f) 21t5 + 14t4 + 7t

2. Factor fully.

(a) 20(a+ b) + 14(a+ b)

(b) 12x(x+ y)− 3y(x+ y)

(c) (3x+ y)(x+ y)− 3y(x+ y)

(d) 10x2(a+ b)− 4x(a+ b)

(e) x7 − x6 + x5 − x4

(f) 49t6 − 14t5

(g) 7t(x+ 2y)− 84(x+ 2y)

(h) 2a(a− b) + b(a− b)

3. Factor fully.

(a) ax+ by + bx+ ay

(b) ax+ bx− ay − by

(c) x3 + x− 3x4 − 3x2

(d) 15ab+ 3b2 − 10a2 − 2ab

(e) r − a+ (r − a)2

(f) 1 + z + z2 + z3


4. Factor fully.

(a) a2 + ab− a+ ab+ b2 − b

(b) p5 − p4 + p3 − p2 + p− 1

(c) abx2 + (an+ bm)x+mn

(d) m(1− z)−mz +mz2

(e) q2a2 + 2q2 + 27a− 9a2 − 3q2a− 18

(f) 7r3 − 3r2s+ 14rs− 6s2 + 8s+ 4r2

Exercises 1.5: Factoring Quadratic and Spe-cial Product Expressions

1. Factor fully.

(a) x2 + 3x+ 2

(b) x2 + 7x+ 12

(c) x2 + x− 30

(d) x2 − 6x− 40

(e) x2 + 5x− 24

(f) x2 − 14x+ 49

(g) y2 + 2y − 24

(h) a2 − 2ab− 15b2

(i) t6 − 4t3 − 12

2. Factor fully.

(a) 2x2 + 5x+ 2

(b) 12x2 − x− 1

(c) 10x2 − 39x− 27

(d) 2x2 − 5x− 12

(e) 6a2 + a− 12

(f) 6x2 + 11xy + 4y2

(g) 5m2 − 24mn− 5n2

(h) a2 + 2 +1

a2

(i) 16h2 − 9hk − 25k2

3. Determine integer values for a and b if possible.

(a) a+ b = 3, ab = −10


(b) a+ b = 5, ab = −36

(c) a+ b = 10, ab = −24

(d) a+ b = 11, ab = 30

(e) a+ b = 13, ab = −30

(f) a+ b = 22, ab = 121

4. Factor fully.

(a) (x+ a)2 − 25

(b) 4− 64a2

(c) x4 − 64

(d) 9(x+ 2y + z)2 − 16(x− 2y + z)2

(e) 36(2x− y)2 − 25(u− 2y)2

(f) 8u2(u+ 1) + 2u(u+ 1)− 3(u+ 1)

(g) 27x2 − 48

(h) a2 + 2ab+ b2 − 1

5. Factor fully.

(a) 125a3 − 27b3

(b) 125a3 + 27b3

(c) z6 − 1

(d) z12 − 1

(e) 343a6b3 − 125

6. For the polynomials f(x) = x2 + ax+ b and g(x) =x3 + ax2 + bx + c, express the coefficients a, b andc in terms of the factors p, q and r, where f(x) =(x− p)(x− q) and g(x) = (x− p)(x− q)(x− r).

Exercises 1.6: Graphing Polynomials

1. Graph the following polynomials. For each poly-nomial, state the domain, range, x-intercepts andy-intercepts.

(a) y = 3(x− 1)(x− 2)(x− 3)

(b) y = (x− 2)(x+ 1)(x− 2)

(c) y = 5(x− 1)(x− 1)(x+ 3)(x+ 4)

(d) y = (x− 7)(x− 5)(x+ 3)(x+ 5)

(e) y = −3(x− 1)(x− 1)

(f) y = −(x− 1)(x− 1)(x− 1)

(g) y = 4(x− 1)(x+ 1)(x− 1)(x+ 1)

(h) y = −3(x− 1)(x− 1)(x− 1)(x− 1)

2. Without using a graphing calculator, determine thebehaviour of the following functions for large pos-itive and negative x-values. Determine the x andy-intercepts for each function.

(a) y = (x+ 1)(x− 1)

(b) y = 2(x+ 1)(x+ 1)(x+ 1)

(c) y = −3(x− 1)(x+ 1)(x− 1)(x− 1)

(d) y = 4(x− 5)(x+ 4)(x− 2)(x− 5)(x− 5)

(x− 5)

(e) y = −(x− 1)(x+ 1)(x+ 1)(x− 2)(x− 3)

(f) y = −2(x+ 1)(x− 1)(x− 1)(x− 1)(x− 1)

(x+ 1)(x− 1)

(g) y = 3(x+ 1)(x− 1)(x− 1)(x− 1)(x− 1)

(h) y = −(x− 1)(x+ 2)(x− 3)(x+ 1)(x+ 2)


(x− 3)(x− 4)(x− 2)

(i) y = 3(x− 2)(x− 2)(x+ 7)(x+ 2)(x− 2)

(x+ 3)(x− 4)

3. Plot the functions y = xn for n = 2, 4, 6, 8, 10 onthe same graph and n = 1, 3, 5, 7, 9 on another forthe domain D = x ∈ R| − 2 ≤ x ≤ 2. Summarizeyour findings.

4. Determine which parts of the domains in the func-tions in Exercise #2 are positive or negative by eval-uating test points between the x-intercepts.

5. Graph each function in Exercise #2.

Exercises 1.7: Translations of Polynomials

1. Plot the following functions on the same graph.

(a) y = x3

(b) y = x3 − 1

(c) y = x3 + 1

(d) y = x3 + 2

(e) y = x3 − 2

(f) y = x3 − 3


(a) y = x4

(b) y = x4 − 1

(c) y = x4 + 1

(d) y = x4 + 2

(e) y = x4 − 2

(f) y = x4 − 3

3. Describe how the graphs of y = x3+c and y = x4+cchange with different values of c.


(a) y = x3

(b) y = (x− 1)3

(c) y = (x+ 1)3

(d) y = (x+ 2)3

(e) y = (x− 2)3

(f) y = (x− 3)3



(a) y = x4

(b) y = (x− 1)4

(c) y = (x+ 1)4

(d) y = (x+ 2)4

(e) y = (x− 2)4

(f) y = (x− 3)4

6. Describe how the graphs of y = (x − d)3 and y =(x− d)4 change with different values of d.

7. Find the equation of the polynomial based on y =x3 that has been translated up by 3.

8. Find the equation of the polynomial based on y =x4 that has been translated left by 4.

9. Plot the following functions.

(a) y = (x− 1)3 + 4

(b) y = (x+ 2)4 − 3

(c) y = (x− 1)4 + 3

(d) y = (x− 2)3 − 5

10. Plot the functions y = 5x and y = x3 on the samegraph. Combine the two graphs to plot y = x3−5x.Now graph the function y = x3 − 3x2 − 2x + 6 bynoting that (x−1)3−5(x−1)+2 = x3−3x2−2x+6.

11. Show that there is a change of variables of the forms = x + k that transforms the general cubic ax3 +bx2 + cx+ d to the form as3 +ms+ n. Use this toconsider the graph of the general cubic.

12. Find a translation of x-coordinates that eliminatesthe cubic term in the general quartic polynomial.

13. Is it always possible to translate coordinates so thatan n-th degree polynomial can have the (n − 1)-thdegree term eliminated? Prove or disprove.

Exercises 1.8: Scaling Polynomials


(a) y = x3

(b) y = 2x3

(c) y = −2x3

(d) y = −x3

(e) y =1

2x3

(f) y = −1

2x3


(a) y = x4

(b) y = 2x4

(c) y = −2x4

(d) y = −x4

(e) y =1

2x4

(f) y = −1

2x4

3. Describe how the graphs of y = ax3 and y = ax4

change with different values of a.


(a) y = x3

(b) y = (2x)3

(c) y = (−x)3

(d) y = (−2x)3

(e) y =

(1

2x

)3


(f) y =

(−1

2x

)3


(a) y = x4

(b) y = (2x)4

(c) y = (−x)4

(d) y = (−2x)4

(e) y =

(1

2x

)4

(f) y =

(−1

2x

)4

6. Describe how the graphs of y = (kx)3 and y = (kx)4

change with different values of k.

7. Plot the following functions over a suitable domain.

(a) y = 3(x− 2)3 + 1

(b) y = 4(x+ 1)4 − 3

(c) y =1

5(2(x+ 1))4 − 5

(d) y = 2(x

3− 1)3− 2

Exercises 1.9: Finding Equations of Polyno-mials

1. Find an equation for a cubic polynomial that goesthrough the intercepts given.

(a) 1,−1, 5

(b) 0, 3, 6

(c) −1, 2, 4

2. Find an equation for a 3rd degree polynomial withx-intercepts 0, 1, 2. Sketch a graph of this function.

3. What are all possible polynomials that have just theintercepts given in Exercise #2?

4. Find an equation of the cubic polynomial thatpasses through (1, 5) and has x-intercepts 0, 2 and7.

5. Find an equation of the quartic polynomial thatpasses through (0, 8) and has x-intercepts -3, 2, 3and 6.

6. Find an equation of the quartic polynomial thatpasses through (2, 3) and has x-intercepts -6, -5, -4and 8.

7. Find an equation of the quartic polynomial thatpasses through (0, 1) and has x-intercepts -2, 1, 3and 4.

8. Find an equation of the cubic polynomial thatpasses through (5, 35) and has x-intercepts 0, 2 and9.

9. Find all equations of the cubic polynomial thatpasses through (−11, 2) and has x-intercepts 1 and8.

10. Find all equations of the quartic polynomial thatpasses through (1, 1) and has x-intercepts 2, 3 and-1.

11. Find all possible equations of a quintic polynomialthat has x-intercepts at 0 and 1.


Exercises 1.10: (*) Even and Odd Polynomi-als

1. Determine which of the following polynomials areeven, odd or neither using the transformation x 7→−x.

(a) y = x4

(b) y = x7

(c) y = x7 + 4x3

(d) y = 3x6 + πx4

(e) y = 5x4 + 3x3 − x2

(f) y = x5 − 2x3 + x

(g) y = 17x5 − 7x4 + x2

(h) y = x2k+1, k ∈ N

(i) y = x2k + 1, k ∈ N

2. State a general rule under which a polynomial y =∑nk=0 akx

k is even, odd or neither.

3. Find examples of even quartic polynomials thathave 0, 1, 2, 3 or 4 x-intercepts.

(a) Determine conditions for which even poly-nomials have an even or odd number of x-intercepts.

(b) Is there a corresponding set of conditions forodd polynomials?

4. For each of the following polynomials, determine aline of symmetry and if the polynomial is even, oddor neither around that line.

(a) y = (x− 1)5 + (x− 1)3

(b) y = (x+ 3)5 − 7(x+ 3)

(c) y = x2 + 3x+ 2

(d) y =(x2 − 4

)2+ x2 + 2

5. Is it possible for a polynomial to be even aroundtwo or more lines of symmetry? Construct such apolynomial or prove that it is not possible.

6. Given two polynomials f(x) and g(x), which of thefollowing functions are even, odd or neither?

(a) f(x) + g(x), if f , g are even.

(b) f(x)− g(x), if f , g are odd.

(c) f(x) + g(x), if f is even, g is odd.

(d) f(x)− g(x), if f is even, g is odd.

(e) f(x)g(x), if f , g are even.

(f) f(x)g(x), if f , g are odd.

(g) f(x)g(x), if f is even, g is odd.

(h) f(g(x)), if f , g are even.

(i) f(g(x)), if f , g are odd.

(j) f(g(x)), if f is even, g is odd.

(k) f(g(x)), if f is odd, g is even.


Exercises 1.11: Review

1. For each polynomial, state the degree, domain,range and leading coefficient.

(a) 7x4 + 2x2 + 3

(b) 8

(c) x5 + 3x2 − 7x− 2

(d) xn + n(n− 1)xn−2 + 3, n ≥ 3

2. Determine the behaviour of each polynomial forlarge positive and large negative x-values.

(a) y = −2x7 + x4 + 3x+ 1

(b) y = x5 − x+ 1

(c) y = x2n

(d) y = x2n+1

3. Sketch all possible graphs of quartic polynomials.

4. Categorize all possible ranges for polynomials.

5. Provide two reasons why y = tanx is not a polyno-mial.

6. Determine the equation of a polynomial using finitedifferences on the data set provided.

(a) (1, 5), (2, 75), (3, 383), (4, 1223),

(5, 3009), (6, 6275)

(b) (1,−7), (2,−16), (3,−9), (4, 32),

(5, 125), (6, 288)

(c) (1, 3), (2,−5), (3,−25), (4,−63),

(5,−125), (6,−217)

7. Factor fully.

(a) 7x4 − 21x2

(b) 3x+ 36

(c) 4ax+ 16bx

(d) x8 − x6 + x4 − x2

(e) 5a(b+ c)− 2d(b+ c)

(f) (x+ a) + (x+ a)2

(g) 3mn− 3m2 + n2 − nm

(h) yz2 − z − zy + y2

8. Factor fully.

(a) x2 − x− 12

(b) x2 + 7x+ 10

(c) x2 + 11x+ 24

(d) 2x2 + 5x+ 3

(e) 2x2 − 5x+ 3

(f) 8x2 + 22x+ 15

9. Factor fully.

(a) (x+ 3)2 − 36

(b) (x− 2)2 − (x+ 3)2

(c) a2 + b2 + c2 + 2ab+ 2ac+ 2bc

(d) x2 + 2x+ 1− (a2 + 2ax+ x2)

10. Graph the following polynomials. For each func-tion, find the domain and range, the x and y-intercepts and the behaviour for large positive and


large negative x-values.

(a) y = 2(x+ 1)(x− 1)(x− 3)

(b) y = −3(x+ 5)(x+ 3)(x+ 1)(x− 1)

(c) y = −6(x+ 2)2(x− 3)

(d) y = 7(x− 1)5(x+ 1)

11. Graph the following functions.

(a) y = x3 + 1

(b) y = x4 − 5

(c) y = (x+ 1)3

(d) y = (x− 2)4

(e) y = (x+ 3)4 + 1

(f) y = (x− 7)3 − 2

12. Graph the following functions.

(a) y = 3x3

(b) y = 4(x− 2)4 + 5

(c) y = −1

2(x+ 1)4 − 1

(d) y = −(−x− 1)4 + 3

(e) y = 4(x− 3)4 − 5

(f) y = −2(3− x)3 + 1

13. Find an equation of a cubic polynomial that passesthrough (1, 3) and has x-intercepts 5, 6 and 7.

14. Find an equation of a quartic polynomial thatpasses through (2, 2) and has x-intercepts -2, -1,0 and 1.

15. Which of the following polynomials are even, oddor neither?

(a) y = x3 + x

(b) y = x4 − 7x2 + 1

(c) y = x7 + x6 + 5x

(d) y = −2x4 + x2


Exercises 1.1: Solutions Algebra Review

1. Simplifying.

(a) −2x2 + 42x− 33

(b) −3a2 + 16b2 + 9ab

(c) x6 − 7x5 + 24x4 − 28x3 + 24x2 + x+ 1

(d)2x+ 17y

24

(e)3x2 + 2xy + 3y2

2x2 − 2y2

(f)a2 + 2ab+ b2 + a4 − 4a3b+ 6a2b2 − 4ab3 + b4

a3 − a2b− ab2 + b3

(g)2a

a2 − b2

2. Simplifying.

(a)a3 + 4a2 − 2a− 1

a3 − a)

(b)4a2b2

b4 − a4

(c) −(a2 − b2

)22ab (a2 + b2)2

3. Substitution.

(a)3b2(a+ b)2

a2

(b) (2a)2b

(c) (2a)a/b

Exercises 1.2: Solutions Definition of Poly-nomials

1. Polynomial definitions.

(a) degree 2, quadratic

(b) degree 4, quartic

(c) degree 1, linear

(d) degree n

(e) not a polynomial

(f) not a polynomial

(g) degree 2, quadratic

(h) only a polynomial if n is positive integer

(i) not a polynomial

(j) degree 0, constant

(k) not a polynomial

(l) not a polynomial

2. Domain, range and behaviour at infinity.

(a) D = R, R = R, as x→ ±∞, y → ±∞

(b) D = R, R = {y ∈ R|y ≥ 7}, y →∞

(c) D = R, R = R, as x→ ±∞, y → ∓∞

(d) D = R, R = {y ∈ R|y ≥ −12.25}, y →∞

(e) D = R, R = R, as x→ ±∞, y → ±∞

(f) D = R, R = R, as x→ ±∞, y → ±∞

(g) D = R, R = R, as x→ ±∞, y → ∓∞

(h) D = R, R = R, as x→ ±∞, y → ±∞


3. Graphs.

(a) straight lines

(b) single global maximum or minimum

(c) local min and max or single inflexion point

(d) single global max or min; two local max; twolocal min; one inflexion and one global max ormin

4. Behaviour at infinity.

(a) as x→ ±∞, y → ±∞(b) as x→ ±∞, y → ∓∞

5. There are an infinite number of zeros, and the func-tion does not approach positive or negative infinity.

6. y1(0) = 6, y2(0) = −6

7. 116037611753629924198558475

8. Look at leading terms of f(x) and f(x + k). Sub-tract the two polynomials, which forces the leadingcoefficient to be zero, which is a contradiction. Thisshows that polynomials can never be periodic.

Exercises 1.3: Solutions Finite Differencesfor Polynomials

1. Finding polynomials.

(a) x3 − 2x2 − 5x+ 3

(b) −2

3x3 − 5

3x− 2

(c)2

3x3 − x2 +

4

3x

(d) −3x4 − x3 + x+ 5

(e) 2x3 + 3x2 − x+ 4

(f) −8x2 + 7x+ 3

(g) −7x4 + 7x3 + 11x+ 3

2. Difference operators.

(a) ∆2f(x) = f(x+ 2)− 2f(x+ 1) + f(x).

3. Difference operators and series.

(a) With f(x) = x2 and a = 1, then S = 1 +4 + 9 + · · · + n2. Also, using the differenceformula, then S = n+ n(n−1)

2 3+ n(n−1)(n−2)6 2 =

n(n+1)(2n+1)6 .

(b) With f(x) = x3 and a = 1, then S = 1 +8 + 27 + · · · + n3. Also, using the differenceformula, then S = n+n(n−1)

2 7+n(n−1)(n−2)6 12+

n(n−1)(n−2)(n−3)24 6 =

(n(n+1)

2

)2.

(c) 3 + 8 + 15 + · · ·+ n(n+ 2) = n(n+1)(2n+7)6 .


Exercises 1.4: Solutions Common Factors andFactoring by Grouping

1. Factoring.

(a) 3x(2x2 + 3)

(b) 5(x+ 5)

(c) x(11y + 4)

(d) 50r

(e) 2m(22n+m2)

(f) 7t(3t4 + 2t3 + 1

2. Factoring.

(a) 34(a+ b)

(b) 3(4x− y)(x+ y)

(c) (3x− 2y)(x+ y)

(d) 2x(5x− 2)(a+ b)

(e) x4(x− 1)(x2 + 1)

(f) 7t5(7t− 2)

(g) 7(x+ 2y)(t− 12)

(h) (2a+ b)(a− b)

3. Factoring.

(a) (a+ b)(x+ y)

(b) (a+ b)(x− y)

(c) x(1− 3x)(x2 + 1)

(d) (3b− 2a)(5a+ b)

(e) (r − a)(1 + r − a)

(f) (1 + z)(1 + z2)

4. Factoring.

(a) (a+ b)(a+ b− 1)

(b) (p− 1)(p2 + p+ 1)(p2 − p+ 1)

(c) (ax+m)(bx+ n)

(d) m(1− z)2

(e) (q + 3)(q − 3)(a− 2)(a− 1)

(f) (r2 + 2s)(7r − 3s+ 4)


Exercises 1.5: Solutions Factoring Quadraticand Special Product Expressions

1. Factoring.

(a) (x+ 1)(x+ 2)

(b) (x+ 3)(x+ 4)

(c) (x+ 6)(x− 5)

(d) (x− 10)(x+ 4)

(e) (x+ 8)(x− 3)

(f) (x− 7)2

(g) (x+ 6)(x− 4)

(h) (a− 5b)(a+ 3b)

(i) (t3 − 6)(t3 + 2)

2. Factoring.

(a) (2x+ 1)(x+ 2)

(b) (4x+ 1)(3x− 1)

(c) (2x− 9)(5x+ 3)

(d) (2x+ 3)(x− 4)

(e) (3a− 4)(2a+ 3)

(f) (2x+ y)(3x+ 4y)

(g) (5m+ n)(m− 5n)

(h)

(a+

1

a

)2

(i) (h+ k)(16h− 25k)

3. Solving.

(a) a = −2, b = 5

(b) a = −4, b = 9

(c) a = −2, b = 12

(d) a = 5, b = 6

(e) a = −2, b = 15

(f) a = 11, b = 11

4. Factoring.

(a) (x+ a− 5)(x+ a+ 5)

(b) 4(1− 4a)(1 + 4a)

(c) (x+√

8)(x−√

8)(x2 + 8)

(d) (−x+ 14y − z)(7x− 2y + 7z)

(e) (12x− 5u+ 4y)(12x+ 5u− 16y)

(f) (u+ 1)(2u− 1)(4u+ 3)

(g) 3(3x− 4)(3x+ 4)

(h) (a+ b+ 1)(a+ b− 1)

5. Factoring.

(a) (5a− 3b)(102 + 15ab+ 9b2

)(b) (5a+ 3b)

(102 − 15ab+ 9b2

)(c) (z − 1)(z + 1)

(z2 − z + 1

) (z2 + z + 1

)(d) (z − 1)(z + 1)

(z2 − z + 1

) (z2 + z + 1

)(z2 + 1

) (z4 − z2 + 1

)(e)

(7a2b− 5

) (49a4b2 + 35a2b+ 25

)6. For f(x) = x2+ax+b, then a = −(p+q) and b = pq.

For g(x) = x3 +ax2 + bx+ c, then a = −(p+ q+ r),b = pq + rq + rp and c = −pqr.


Exercises 1.6: Solutions Graphing Polynomi-als

1. Graphing polynomials and key features.

(a) D = R, R = R, x-int at x = 1, 2, 3, y-int aty = −18.

(b) D = R, R = R, x-int at x = 2, −1, y-int aty = 4.

(c) D = R, R = {y ∈ R|y ≥ −25.6}, x-int atx = 1, −4, −3, y-int at y = 60.

(d) D = R, R = {y ∈ R|y ≥ −99.5}, x-int atx = 7, 5, −3, −5, y-int at y = 525.

(e) D = R, R = {y ∈ R|y ≤ 0}, x-int at x = 1,y-int at y = −3.

(f) D = R, R = R, x-int at x = 1, y-int at y = 1.

(g) D = R, R = {y ∈ R|y ≥ 0}, x-int at x = 1,−1, y-int at y = 4.

(h) D = R, R = {y ∈ R|y ≤ 0}, x-int at x = 1,y-int at y = −3.

2. Behaviour at infinity and intercepts.

(a) As x→∞, y →∞, as x→ −∞, y →∞, x-intat x = −1, 1, y-int at y = −1.

(b) As x → ∞, y → ∞, as x → −∞, y → −∞,x-int at x = −1, y-int at y = 2.

(c) As x → ∞, y → −∞, as x → −∞, y → −∞,x-int at x = −1, 1, y-int at y = 3.

(d) As x → ∞, y → ∞, as x → −∞, y → −∞,x-int at x = 5, −4, 2, y-int at y = −20000.

(e) As x → ∞, y → −∞, as x → −∞, y → ∞,x-int at x = −1, 1, 2, 3, y-int at y = 6.

(f) As x → ∞, y → −∞, as x → −∞, y → ∞,x-int at x = −1, 1, y-int at y = 2.

(g) As x → ∞, y → ∞, as x → −∞, y → −∞,x-int at x = −1, 1, y-int at y = 3.

(h) As x → ∞, y → −∞, as x → −∞, y → −∞,x-int at x = −1, 1, −2, 2, 3, 4, y-int at y =−288.

(i) As x→∞, y →∞, as x→ −∞, y → −∞, x-int at x = 2, −2, −3, 4, −7, y-int at y = 4032.

Exercises 1.7: Solutions Translations ofPolynomials

1. Plotting cubics.

2

1 2 3−1−2−3

4

6

8

10

−2

−4

−6

−8

−10

2. Plotting quartics.

2

1 2 3−1−2−3

4

6

8

10

−2

−4

12

14

16

18

3. If c ≥, then the graph shifts up by c. If c < 0, thenthe graph shifts down by −c.


4. Horizontal cubic translation.

2

1 2 3 4−1−2−3

4

6

8

−4−2

−4

−6

−8

5

5. Horizontal quartic translation.

2

1 2 3 4−1−2−3

4

6

8

10

−25

12

14

16

−4

6. If d ≥ 0, then the graph shifts right by d. If d < 0,then the graph shifts left by −d.

7. y = x3 + 3.

8. y = (x− 4)4.

9. Horizontal and vertical translations.

2

1 2 3 4−1−2−3

4

6

8

10

−2

−4

12

14

16

18

−4

20

−6

−8

−10

−12

10. The graphs will be the same.

11. Let s = x+ b/3a to eliminate the quadratic term.

12. Let s = x+ b/4a to eliminate the cubic term.

13. Yes. Let s = x+ b/na where n is the degree of thepolynomial.


Exercises 1.8: Solutions Scaling Polynomials

1. Vertically scaling cubics.

1

1 2 3−1−2−3

2

3

4

5

−1

−2

6

7

8

−7

−8

−3

−4

−5

−6

2. Vertically scaling quartics.

2

1 2 3−1−2−3

4

6

8

10

−2

−3

12

14

16

−14

−16

−6

−7

−10

−12

3. If a > 1, vertical stretch by a; if 0 < a < 1, ver-tical compression by 1/a; if −1 < a < 0, verticalreflection, compression by −1/a; if a < −1, verticalreflection, stretch by −a.

4. Horizontally scaling cubics.

1

1 2 3 4−1−2−3

2

3

4

5

−1

−2

6

7

8

−7

−4

−8

−3

−4

−5

−6

5. Horizontally scaling quartics.

2

1 2 3 4−1−2−3

4

6

8

10

12

14

16

−4

6. Expand the functions to y = k3x3 and y = k4x4,then use appropriate vertical scaling.


7. Transformed cubics and quartics.

1

1 2 3 4−1−2

2

3

4

−1

−2

5 6 7

−3

−4

−5

Exercises 1.9: Solutions Finding Equations ofPolynomials

1. Cubic polynomials and intercepts.

(a) y = (x− 1)(x+ 1)(x− 5).

(b) y = x(x− 3)(x− 6).

(c) y = (x+ 1)(x− 2)(x− 4).

2. y = x(x− 1)(x− 2).

3. y = kx(x− 1)(x− 2).

4. y = 56x(x− 2)(x− 7).

5. y = − 227(x+ 3)(x− 2)(x− 3)(x− 6).

6. y = − 1672(x+ 6)(x+ 5)(x+ 4)(x− 8).

7. y = − 124(x+ 2)(x− 1)(x− 3)(x− 4).

8. y = − 712x(x− 2)(x− 9).

9. y = − 11368(x− 1)2(x− 8) and y = − 1

2166(x− 1)(x−8)2.

10. y = −14(x − 2)2(x − 3)(x + 1), y = −1

8(x − 2)(x −3)2(x+ 1) and y = 1

8(x− 2)(x− 3)(x+ 1)2.

11. y = kx(x − 1)4, y = kx2(x − 1)3, y = kx3(x − 1)2

and y = kx4(x− 1),


Exercises 1.10: Solutions Even and Odd Poly-nomials

1. Even and odd polynomials.

(a) Even.

(b) Odd.

(c) Odd.

(d) Even.

(e) Odd.

(f) Even.

(g) Neither.

(h) Odd.

(i) Even.

2. If all powers of x are even, the polynomial is even,if all powers of x are odd, the polynomial is odd,otherwise, the polynomial is neither even nor odd.

3. y = x4 + 1 has 0 intercepts, y = x4 has 1 intercept,y = (x − 1)2(x + 1)2 has 2 intercepts, y = (x −1)2(x+ 1)2− 1 has 3 intercepts and y = (x− 1)(x−2)(x+ 1)(x+ 2) has 4 intercepts.

(a) Even polynomials have an even number ofx-intercepts if the intercepts are symmetricaround the x-axis, so the polynomial is of theform y =

∑ni=0 ai(x − bi)

qi(x + bi)qi . Even

polynomials will have an odd number of x-intercepts if, in addition to the previous con-dition, the y-intercept is 0. The polynomialis now of the form y =

∑ni=0 ai(x − bi)qi(x +

bi)qi + k, where k = −2

∑ni=0 ai(bi)

qi .

(b) Odd polynomials always have an odd numberof x-intercepts.

4. Lines of symmetry.

(a) x = 1, odd.

(b) x = −3, odd.

(c) x = −3/2, even.

(d) x = 0, even.

5. Not possible. Lines of symmetry must be aroundthe “central” point of a function.

6. Even, odd or neither?

(a) Even.

(b) Odd.

(c) Neither.

(d) Neither.

(e) Even.

(f) Even.

(g) Odd.

(h) Even.

(i) Odd.

(j) Even.

(k) Even.


Exercises 1.11: Solutions Polynomial Review

1. Classifying polynomials.

(a) Deg 4, D = R, R = {y ∈ R|y ≥ 3}, l.c. 7.

(b) Deg 0, D = R, R = {y = 8}, l.c. 8.

(c) Deg 5, D = R, R = R, l.c. 1.

(d) Deg n, D = R, R = {y ∈ R|y ≥ 3, n odd,y ∈ R, n even}, l.c. 1.

2. Behaviour at infinity.

(a) As x→ ±∞, y → ∓∞.

(b) As x→ ±∞, y → ±∞.

(c) As x→ ±∞, y →∞.

(d) As x→ ±∞, y → ±∞.

3. See earlier graphs.

4. Ranges are either all real numbers, all real num-bers greater than the global minimum, or all realnumbers less than the global maximum.

5. The domain is not all real numbers and there arean infinite number of x-intercepts.

6. Finite differences.

(a) y = 5x4 − x3 + 2x− 1.

(b) y = 3x3 − 10x2.

(c) y = −x3 − x+ 5.

7. Factoring.

(a) 7x2(x−√

3)(x+√

3)

(b) 3(x+ 12)

(c) 4x(a+ 4b)

(d) x(x+ 1)(x− 1)(x5 + 1)

(e) (b+ c)(5a− 2d)

(f) (x+ a)(1 + x+ a)

(g) (n−m)(3m+ n)

(h) Does not factor.

8. Factoring.

(a) (x− 4)(x+ 3)

(b) (x+ 5)(x+ 2)

(c) (x+ 8)(x+ 3)

(d) (2x+ 3)(x+ 1)

(e) (2x+ 1)(x− 3)

(f) (4x+ 5)(2x+ 3)

9. Factoring.

(a) (x+ 9)(x− 3)

(b) −5(2x+ 1)

(c) (a+ b+ c)2

(d) (1− a)(2x+ a+ 1)

10. Graphing.

(a) D = R, R = R, x-int at -1, 1, 3, y-int= 6, asx→ ±∞, y → ±∞

(b) D = R, R = {y ∈ R|x ≤ 48}, x-int at -5, -3,-1, 1, y-int= 45, as x→ ±∞, y → −∞

(c) D = R, R = R, x-int at -2, 3, y-int= 72, asx→ ±∞, y → ∓∞

(d) D = R, R = {y ∈ R|x ≥ −30}, x-int at -1, 1,y-int= −7, as x→ ±∞, y →∞

1

1 2 3 4−1−2−3

2

3

4

−1

−2

−3

−4−5−6

11. Transformed polynomials.

(a) Vertical shift up 1.

(b) Vertical shift down 5.

(c) Horizontal shift left 1.

(d) Horizontal shift right 2.

(e) Horizontal shift left 3, vertical shift up 1.


(f) Horizontal shift right 7, vertical shift down 2.

12. Transformed polynomials.

(a) Vertical stretch by factor of 3.

(b) Vertical stretch by factor of 3, vertical shift up5.

(c) Vertical reflection, vertical compression by fac-tor of 2, horizontal shift left 1, vertical shiftdown 1.

(d) Vertical reflection, horizontal shift left 1, ver-tical shift up 3.

(e) Vertical stretch by factor of 4, horizontal shiftright 3, vertical shift down 5.

(f) Vertical reflection, vertical stretch by factor of3, horizontal reflection, horizontal shift right3, vertical shift up 1.

13. y = − 140(x− 5)(x− 6)(x− 7)

14. y = 112x(x+ 2)(x+ 1)(x− 1)

15. Even, odd or neither.

(a) Odd.

(b) Even.

(c) Neither.

(d) Even.

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Exercises 1.2: Definition of Polynomials b a b a b a b x …€¦Unit 1 Polynomials Exercises MHF 4UI Page 1 Exercises 1.1: Algebra Review 1. Simplify. ... poly-nomial if the x5 term