Chapter 9 Final Exam Review

Add Polynomials (2x² + x³ – 1) (2x² + x³ – 1) Like Terms terms that have the same variable (2x³ – 5x² + x) + (2x³ – 5x² + x) + You can add polynomials using the vertical or horizontal format. Vertical Format 2x³ – 5x² + x 2x³ – 5x² + x x³ + 2x² – 1 x³ + 2x² – 1 3x³ – 3x² + x – 1 3x³ – 3x² + x – 1 Horizontal Format (2x³ + x³) + (2x² – 5x²) + x – 1 (2x³ + x³) + (2x² – 5x²) + x – 1 3x³ – 3x² + x – 1 3x³ – 3x² + x – 1

Subtract Polynomials (-2n² + 2n – 4) (-2n² + 2n – 4) Like Terms terms that have the same variable (4n² + 5) – (4n² + 5) – You can subtract polynomials using the vertical or horizontal format. Vertical Format 4n² + 5 4n² + 5 – (-2n² +2n – 4) – (-2n² +2n – 4) Horizontal Format (4n² + 2n²) – 2n + (5 + 4) (4n² + 2n²) – 2n + (5 + 4) 6n² – 2n + 9 6n² – 2n + 9 +(2n² -2n + 4) +(2n² -2n + 4) 6n² – 2n + 9 6n² – 2n + 9

Section 9.2 “Multiply Polynomials” When multiplying polynomials use the distributive property. Distribute and multiply each term of the polynomials. Then simply. 2x³ (x³ + 3x² - 2x + 5)

“Multiply Using FOIL” When multiplying a binomial and another polynomial use the method. FOILFOIL FirstOuterInnerLast

(3a + 4) (a – 2) “Multiply Using FOIL” combine like terms

Section 9.4 “Solve Polynomial Equations in Factored Form” If ab = 0, then a = 0 or b = 0. The zero-product property is used to solve an equation when one side of the equation is ZERO and the other side is the product of polynomial factors. (x – 4)(x + 2) = 0 Zero-Product Property The solutions of such an equation are called ROOTS. x – 4 = 0 x + 2 = 0 x = 4 x = -2

“Solving Equations By Factoring” Look for common terms 2x² + 8x = 0 When using the zero-product property, sometimes you may need to factor the polynomial, or write it as a product of other polynomials. Look for the greatest common factor (GCF) of the polynomial’s terms. GCF- the monomial that divides evenly into EACH term of the polynomial. GCF

Solve Equations By Factoring 2x = 0 x + 4 = 0 x = 0 x = -4 2x² + 8x = 0 2x(x + 4) = 0 Factor left side of equation Zero product property The solutions of the equation are 0 and -4.

Section 9.5 “Factor x² + bx + c” x² + bx + c = (x + p)(x + q) provided p + q = b and pq = c Factoring x² + bx + c x² + 5x + 6 = (x + 3)(x + 2) Remember FOIL

Factoring polynomials ‘-’ factors of 8Sum of factors -8, (-1)= -9 -2, (-4) = -6 n² – 6n + 8 (n – 2)(n – 4) Find two ‘negative’ factors of 8 whose sum is -6.

Factors of 2Factors of 3Possible factorization Middle term when multiplied 1, 21, 3(x – 1)(2x – 3)-3x – 2x = -5x 1, 23, 1(x – 3)(2x – 1)-x – 6x = -7x 2x² – 7x + 3 (x – 3)(2x – 1) Section 9.6 “Factor ax² + bx + c” First look at the signs of b and c.

Factoring Polynomials Completely (1) Factor out greatest common monomial factor. (2) Look for difference of two squares or perfect square trinomial. (3) Factor a trinomial of the form ax² + bx + c into binomial factors. (4) Factor a polynomial with four terms by grouping. 3x² + 6x = 3x(x + 2) x² + 4x + 4 = (x + 2)(x + 2) 16x² – 49 = (4x + 7)(4x – 7) 3x² – 5x – 2 = (3x + 1)(x – 2) -4x² + x + x³ - 4 = (x² + 1)(x – 4)