Answer:
1. (x + 8)(x + 3)
2. (x - 9)²
Step-by-step explanation:
Given trinomials:
1. [tex]x^2+11x+24[/tex]
2. [tex]x^2-18x+81[/tex]
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When factoring trinomials of the form ax² + bx + c:
- Multiply the leading coefficient and the last term.
- Find the product factors that add up to give you the coefficient of the middle term.
- Rewrite the polynomial with those factors replacing the middle term.
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1) x² + 11x + 24
Step 1: Multiply the leading coefficient (1) and the last term (24).
[tex]\implies 1 \times 24 = \boxed{24}[/tex]
Step 2: Find the product factors that sum up to the middle term's coefficient (11).
[tex]\begin{array}{| c | c |}\cline{1-2} \sf Factors\:of\:24 & \sf Sum\:of\:factors\\\cline{1-2} 1, 24\ \| -1, -24 & 25\ \| -25 \\\cline{1-2} 2, 12\ \| -2, -12 & 14\ \| -14 \\ \cline{1-2} 3, 8\ \| -3, -8 & 11\ \| -11 \\\cline{1-2} 4, 6 \ \| -4, -6 & 10\ \| -10 \\\cline{1-2}\end{array}[/tex]
[tex]\implies 3x+8x=11x[/tex]
Step 3: Rewrite the polynomial with those factors, replacing the middle term.
[tex]\implies x^2+3x+8x+24[/tex]
Step 4: Factor by grouping.
[tex]\implies \overbrace{(x^2+3x)}^x+\overbrace{(8x+24)}^8\ \ \textsf{[ Factor out $x$ and $8$. ]}[/tex]
[tex]\implies x\overbrace{(x+3)}^{\textsf {Factor out}}+8(x+3)[/tex]
[tex]\implies \boxed{(x+8)(x+3)}[/tex]
The factored form of the given trinomial is [tex](x+8)(x+3)[/tex].
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2) x² - 18x + 81
Step 1: Multiply the leading coefficient (1) and the last term (81).
[tex]\implies 1 \times 81 = \boxed{81}[/tex]
Step 2: Find the product factors that sum up to the middle term's coefficient (-18).
[tex]\begin{array}{| c | c |}\cline{1-2} \sf Factors\:of\:81 & \sf Sum\:of\:factors\\\cline{1-2} 1, 81\ \| -1, -81 & 81\ \| -81 \\\cline{1-2} 3, 27\ \| -3, -27 & 30\ \| -30 \\ \cline{1-2} 9, 9\ \| -9, -9 & 18\ \| -18 \\\cline{1-2}\end{array}[/tex]
[tex]\implies (-9x)+(-9x)=-18x[/tex]
Step 3: Rewrite the polynomial with those factors, replacing the middle term.
[tex]\implies x^2-9x-9x+81[/tex]
Step 4: Factor by grouping.
[tex]\implies \overbrace{(x^2-9x)}^x+\overbrace{(-9x+81)}^{-9}\ \ \textsf{[ Factor out $x$ and $-9$. ]}[/tex]
[tex]\implies x\overbrace{(x-9)}^{\textsf {Factor out}}-9(x-9)[/tex]
[tex]\implies (x-9)(x-9)[/tex]
[tex]\implies \boxed{(x-9)^2}[/tex]
The factored form of the given trinomial is [tex](x-9)(x-9)\ \textsf{or}\ (x-9)^2[/tex].
