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.)