Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig. Do Now: Aim: How do we solve polynomial equations using factoring? Write the expression (x + 1) (x + 2)(x + 3) as a polynomial in standard form. (x + 1)(x + 2)(x + 3) FOIL first two factors (x 2 + 3x + 2)(x + 3) x 3 + 3x 2 + 3x 2 + 9x + 2x + 6 Multiply by distribution of resulting factors Combine like terms x 3 + 6x 2 + 11x + 6 cubic expression is standard form
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Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig. Do Now: Aim: How do we solve polynomial equations using factoring? Write the expression (x.
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Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
Do Now:
Aim: How do we solve polynomial equations using factoring?
Write the expression (x + 1)(x + 2)(x + 3) as a polynomial in standard form.
(x + 1)(x + 2)(x + 3) FOIL first two factors
(x2 + 3x + 2)(x + 3)
x3 + 3x2 + 3x2 + 9x + 2x + 6
Multiply by distribution of resulting factors
Combine like terms
x3 + 6x2 + 11x + 6
cubic expression is standard form
Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
Solving Polynomials by Factoring
Factor: 2x3 + 10x2 + 12x
GCF
Factor trinomial
2x(x2 + 5x + 6x)
2x(x + 3)(x + 2)
Solve: 2x3 + 10x2 + 12x = 0
2x(x2 + 5x + 6x) = 0
2x(x + 3)(x + 2) = 0
Set factors equal to zero(Zero Product Property)
2x = 0(x + 3) = 0(x + 2) = 0
x = 0, -2, -3
Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
Graphical Solutions
Solve: 2x3 + 10x2 + 12x = 0 2x(x + 3)(x + 2) = 0
x = 0, -2, -3
2
-2
-4
f x = 2x3+10x2+12x
ax3 + bx2 + cx + d = y
Cubic equation in Standard form
y-intercept
Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
4
2
-2
-4
-6
-8
Model Problem
Find the zeros of y = (x – 2)(x + 1)(x + 3)
0 = (x – 2)(x + 1)(x + 3)
Set factors equal to zero(x – 2) = 0(x + 1) = 0(x + 3) = 0
Zeros/roots/x-intercepts are found at y = 0 (x-axis)
x = -3, -1, 2y-intercept?
(-2)(1)(3) = -6
4
2
-2
-4
-6
-8
is the product of last terms of binomial factors
y = x3 + 2x2 – 5x – 6
Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
Synthetic Division & Factors
Divide x3 – x2 + 2 by x + 1 using synthetic division
1 -1 0 2-1
1-1
-2 -2
0
x2 – 2x + 2 0quotient remainder
Since there is no remainder x + 1 is a factor of x3 – x2 + 2
Remainder Theorem
(x2 – 2x + 2)(x + 1) = x3 – x2 + 2
= 0x3 – x2 + 2 Solve: (x2 – 2x + 2)(x + 1) = 0
Set factors equal to zero(x2 – 2x + 2) = 0(x + 1) = 0
x = -1
Quadratic Formula
x = 1 ± i
22
Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
Synthetic Division, Factors and Graphing
= 0x3 – x2 + 2 Solve: (x2 – 2x + 2)(x + 1) = 0
Set factors equal to zero(x2 – 2x + 2) = 0(x + 1) = 0
x = -1
Quadratic Formula
x = 1 ± i
4
3
2
1
-1
-2
-2 2
q x = x3-x2 +2
Aim: Solving Polynomials by Factoring Course: Alg. 2 & Trig.
Model Problem
Use synthetic division to show that x + 3 is a factor of y = 2x3 + 11x2 + 18x + 9 then