Algebra 1 Notes SOL A.2 Type I Factoring (GCF)
Mrs. Grieser
Name: __________________________________________ Date: _____________ Block: _______
Solving Polynomials in Factored Form
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zeros of a function: where a functions value is 0
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x-intercepts: where a function crosses or touches the x-axis (same as zeros!)
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roots: solutions to the equation f(x) = 0
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solutions: values of x that make an equation true
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First degree polynomials (lines!) have one zero.
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Second degree polynomials (called quadratics) can have as many as two zeros.
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When a polynomial function is set to 0, we solve it by finding its roots, which gives us
the zeros of the function.
Zeros of the Product of Two Binomials
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Find the zeros of f(x) = (x + 1)(x – 2) by graphing and find x-intercepts, or…
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Solve (x + 1)(x – 2) = 0.
Find the roots (solutions to the equation)! Use…
The Zero-Product Property
Let a and b be real numbers. If ab = 0, then a = 0 or b = 0.
If (x + 1)(x – 2) = 0, then it must be true that either x + 1 = 0 or x – 2 = 0.
x+1=0
x = -1
x–2=0
x=2
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Therefore, the zeros of f(x) = (x + 1)(x – 2) are -1 and 2.
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Verify by plugging the values back into the original equation.
You try: Solve the equations (find the roots) below…
a) (x – 3)(x + 6) = 0
b) (m – 7)(m – 9) = 0
c) (5n + 10)(4n + 12) = 0
Factoring
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Factor: rewrite a number or expression as a product of primes; e.g. 6 = 2 ∙ 3
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In algebra, factor by rewriting a polynomial as a product of lower-degree polynomials
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What are (x + 1)(x – 2) factors of (multiply!)? _____________________________
Algebra 1 Notes SOL A.2 Type I Factoring (GCF)
Mrs. Grieser Page 2
TYPE I Factoring: “Factor Out” GCF Monomials
Find a common monomial in a polynomial (the GCF) and factor it out (perform reverse
distribution)
Examples: Factor the following polynomials…
a) 6x2 + 12x
b) 12x + 42y
c) 4x4 + 24x3
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GCF = 6x
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GCF = ______
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GCF = ______
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6x(x + 2)
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Factored: __________
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Factored: ___________
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Verify by distributing:
get original polynomial
back!
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Verify!
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Verify!
Solving an Equation by Factoring
When given a polynomial equation to solve,
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make sure all terms are on one side of the equation, and the other side is 0,
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factor the polynomial, then
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find the solutions (roots, or zeros of the function) of the factored version of the
polynomial.
Examples: Solve the polynomials…
a) 2x2 + 8x = 0
b) 9y2 = 21y
2x(x + 4) = 0
9y2 – 21y = 0
c) 12h2 = 36h
Rewrite:___________
2x = 0
x=0
Factor: ___________
Factor: ____________
x+4=0
x = -4
Solve: ___________
Solve: _____________
Verify!
Verify!
Solutions: x= 0, x = -4
Verify!
You try: Solve the equations:
a) (x + 4)(x -5) = 0
b) (2z - 2)(z + 3) = 0
c) (2n – 6)(10n – 5) = 0
e) 9m3 – 3m2
f) 2w2 + 4w
h) 3p2 = 3p
i) -28m2 = 14m
Factor:
d) 4m - 2
Solve the equations:
g) a2 + 8a = 0