Answer
Here are the factored forms of the given expressions:
1. \( x^{2}+3 x+2 = (x+1)(x+2) \)
2. \( x^{2}-10 x+24 = (x-6)(x-4) \)
3. \( 3 x^{2}-15 x+12 = 3(x-4)(x-1) \)
4. \( x^{3}+6 x^{2}+9 x = x(x+3)^{2} \)
5. \( -x^{2}+9 x-20 = -(x-5)(x-4) \)
Solution
Factor the expression by following steps:
- step0: Factor:
\(x^{2}-10x+24\)
- step1: Rewrite the expression:
\(x^{2}+\left(-4-6\right)x+24\)
- step2: Calculate:
\(x^{2}-4x-6x+24\)
- step3: Rewrite the expression:
\(x\times x-x\times 4-6x+6\times 4\)
- step4: Factor the expression:
\(x\left(x-4\right)-6\left(x-4\right)\)
- step5: Factor the expression:
\(\left(x-6\right)\left(x-4\right)\)
Factor the expression \( -x^{2}+9 x-20 \).
Factor the expression by following steps:
- step0: Factor:
\(-x^{2}+9x-20\)
- step1: Rewrite the expression:
\(-x^{2}+\left(4+5\right)x-20\)
- step2: Calculate:
\(-x^{2}+4x+5x-20\)
- step3: Rewrite the expression:
\(-x\times x+x\times 4+5x-5\times 4\)
- step4: Factor the expression:
\(-x\left(x-4\right)+5\left(x-4\right)\)
- step5: Factor the expression:
\(\left(-x+5\right)\left(x-4\right)\)
- step6: Factor the expression:
\(-\left(x-5\right)\left(x-4\right)\)
Factor the expression \( x^{2}+3 x+2 \).
Factor the expression by following steps:
- step0: Factor:
\(x^{2}+3x+2\)
- step1: Rewrite the expression:
\(x^{2}+\left(2+1\right)x+2\)
- step2: Calculate:
\(x^{2}+2x+x+2\)
- step3: Rewrite the expression:
\(x\times x+x\times 2+x+2\)
- step4: Factor the expression:
\(x\left(x+2\right)+x+2\)
- step5: Factor the expression:
\(\left(x+1\right)\left(x+2\right)\)
Factor the expression \( x^{3}+6 x^{2}+9 x \).
Factor the expression by following steps:
- step0: Factor:
\(x^{3}+6x^{2}+9x\)
- step1: Rewrite the expression:
\(x\times x^{2}+x\times 6x+x\times 9\)
- step2: Factor the expression:
\(x\left(x^{2}+6x+9\right)\)
- step3: Factor the expression:
\(x\left(x+3\right)^{2}\)
Factor the expression \( 3 x^{2}-15 x+12 \).
Factor the expression by following steps:
- step0: Factor:
\(3x^{2}-15x+12\)
- step1: Rewrite the expression:
\(3x^{2}-3\times 5x+3\times 4\)
- step2: Factor the expression:
\(3\left(x^{2}-5x+4\right)\)
- step3: Factor the expression:
\(3\left(x-4\right)\left(x-1\right)\)
Here are the factorizations for each of the given expressions:
1. **For \( x^{2}+3 x+2 \)**:
\[
x^{2}+3 x+2 = (x+1)(x+2)
\]
2. **For \( x^{2}-10 x+24 \)**:
\[
x^{2}-10 x+24 = (x-6)(x-4)
\]
3. **For \( 3 x^{2}-15 x+12 \)**:
\[
3 x^{2}-15 x+12 = 3(x-4)(x-1)
\]
4. **For \( x^{3}+6 x^{2}+9 x \)**:
\[
x^{3}+6 x^{2}+9 x = x(x+3)^{2}
\]
5. **For \( -x^{2}+9 x-20 \)**:
\[
-x^{2}+9 x-20 = -(x-5)(x-4)
\]
These factorizations simplify the expressions into products of polynomials.
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