Topics In Applied Mathematics Chapter 13

Factoring Polynomials

MT108

PolynomialsPolynomials are algebraic expressions

containing terms that are added and subtracted

(no negative exponents or variables in the denominator)

A polynomial with one variable is written with the exponents decreasing from left to right.

Types of Polynomials – Monomial: polynomial with one term – Binomial: polynomial with two terms – Trinomial: polynomial with three terms

MT108

Degree of the Polynomial Degree of each term: – sum of the exponents on the variables.

Degree of the polynomial: – the highest degree found on any one term.

Degree of a number (constant term): – zero.

Important vocabulary

• Names for polynomials/terms by degree: – Constant – degree is zero (no variable) – Linear – degree is one – Quadratic – degree is two – Cubic – degree is three

• Leading term: the term with the highest degree when written in descending order – Leading coefficient is the coefficient of the

leading term ☺

MT108

Adding Polynomials • add like terms

Subtracting Polynomials • Distribute the subtraction sign (change all

the signs of the polynomial being subtracted).

• Then combine like terms.

A visual for addition and subtraction

MT108

To evaluate: • Substitute a number for the variable, • Then combine the numbers using the

order of operations.

Evaluate a Polynomial

MT108

Multiplying Polynomials Multiplying a Monomial and a Polynomial • Use distributive property.

Multiplying any two Polynomials • Multiply each term of the 1st polynomial

by each term of the 2nd polynomial. • Then add like terms.

A Visual for Multiplication

MT108

When multiplying two binomials, can remember as the FOIL method.

F – product of First terms O – product of Outside terms

I – product of Inside terms

L – product of Last terms

Then combine like terms

FOIL Method

MT108

When multiplying binomials, there are patterns that lead to special products.

Squaring a Binomial (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2

Multiplying the Sum and Difference of Two Terms (a + b)(a – b) = a2 – b2

Special Products

MT108

Dividing a Polynomial by a Monomial

Divide each term of the polynomial by the monomial:

a+b = a + b (c ≠ 0) c c c

MT108

Dividing a Polynomial by a PolynomialUse long division. Steps of division: D (divide) M (multiply) S (subtract) B(bring

down)

1. Write both polynomials in descending powers. – Add missing exponent terms with coefficient of zero

(e.g., for x3+1, write as x3+0x2+0x+1). 2. Divide using long division. – Write like terms in answer over like terms in the

dividend (dividend is the # being divided).

Examples…

• End Review Section

MT108

Factoring: Common Factor and Difference of Squares

MT108

FactorsFactors: – items multiplied to get a product.

Factoring Polynomials: – reverse of multiplication. – factors will have integer coefficients. – the final factors must be “prime” • have no common factors other than 1 • “Factored Completely”

MT108

Greatest Common Factor

1. Find the greatest common factor (GCF) shared by all the terms, if any.

2. Express each term as the product of the GCF and its other factor.

3. Use the distributive property to factor out the GCF from each term.

4. Check answer using multiplication.

MT108

Difference of Two SquaresA2

– B2 = (A + B)(A - B)

To Factor: • Write the sum and difference of A and B as

products.

• NOTE: A2 + B2 is prime. (Cannot be factored.)

MT108

The Sum and Difference of Cubes

MT108

Sum and Difference of Two Cubes(SOAP)

A3 + B3 = (A + B)(A2 – AB + B2)

Same Sign Always Positive

Opposite Signs

A3 - B3 = (A - B)(A2 + AB + B2)

Same Sign Always Positive

Opposite Signs

MT108

Factoring Trinomials

MT108

Perfect Square Trinomials

A2 + 2AB + B2 = (A + B)2

A2 – 2AB + B2 = (A - B)2

MT108

“Trial and Error”

After removing the GCF,

MT108

By a certain grouping of terms, a common factor is found.

Use when: • have 4 or more terms, • the terms do not contain a common factor.

Factor by Grouping

Factor trinomials by grouping1. Multiply the leading coefficient, 1, and the constant term, c. 1 • (+8) = +8 (Notice: Since the leading coefficient is 1, you can see that the product is = c.) 2. Consider all of the possible factors of this new product. Factors of +8. (1) • (8) (2) • (4) 3. From the list of factors, find the one pair that adds to the middle term's coefficient, b. For this example, we need to find a sum of 6. 2 + 4 = 6

Factoring by grouping …

4. Re-write the middle term, forming two terms, using these two values (order is not important): x2 + 2x +4x +8 5. Group the first two terms together and group the last two terms together. Notice the plus sign between the two groups. (x2 + 2x) + (4x + 8) 6. Factor the greatest common factor out of each group. Watch out for those signs in the second group should a negative be involved. x(x + 2) + 4(x + 2) 7. Notice that the expressions in the parentheses are identical. By factoring out the parentheses binomial, we have the answer: (x + 2)(x +4) ANSWER:

Another example:

•  

Grouping on a Chart

MT108

Strategy for Factoring a Polynomial

1. Factor out the GCF: ab + ac + ad = a(b + c + d). 2. Look at the number of terms.

a) 2 terms: Is it one of these special forms? difference of squares: A2 – B2 = (A + B)(A – B) sum of cubes: A3 + B3 = (A + B)(A2 – AB + B2) difference of cubes: A3 - B3 = (A - B)(A2 + AB + B2)

(Note: A2 + B2 is prime.) b) 3 terms: Is it a perfect square trinomial? A2 + 2AB + B2 = (A + B)2 A2 – 2AB + B2 = (A - B)2

c) 3 terms: Not a perfect square trinomial? ax2 + bx + c !factor by trial and error or factor by grouping method

d) 4 or more terms: Factor by grouping.

Final answer must have prime factors. (No + or – outside parentheses.)