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Unit: Exponents and Polynomials

Match the polynomial expression on the left with the simplified version on the right.

[tex]\[
\frac{6x^3 + 11x^2 - 5x - 12}{3x + 4} \quad 2x^2 + x - 8
\][/tex]

Sagot :

GinnyAnsweridn
To solve the polynomial division [tex]\(\frac{6x^3 + 11x^2 - 5x - 12}{3x + 4}\)[/tex], we will perform a step-by-step polynomial long division.

### Step 1: Setup
Set up the division as you would with numerical long division. The numerator is [tex]\(6x^3 + 11x^2 - 5x - 12\)[/tex] and the denominator is [tex]\(3x + 4\)[/tex].

### Step 2: First Division
Divide the leading term of the numerator, [tex]\(6x^3\)[/tex], by the leading term of the denominator, [tex]\(3x\)[/tex]:
[tex]\[ \frac{6x^3}{3x} = 2x^2 \][/tex]
Write [tex]\(2x^2\)[/tex] as the first term of the quotient.

### Step 3: Multiply and Subtract
Multiply [tex]\(2x^2\)[/tex] by the entire denominator [tex]\(3x + 4\)[/tex]:
[tex]\[ 2x^2 \cdot (3x + 4) = 6x^3 + 8x^2 \][/tex]
Subtract this result from the original numerator:
[tex]\[ (6x^3 + 11x^2 - 5x - 12) - (6x^3 + 8x^2) = 3x^2 - 5x - 12 \][/tex]

### Step 4: Second Division
Divide the new leading term of the numerator, [tex]\(3x^2\)[/tex], by the leading term of the denominator, [tex]\(3x\)[/tex]:
[tex]\[ \frac{3x^2}{3x} = x \][/tex]
Write [tex]\(x\)[/tex] as the next term of the quotient.

### Step 5: Multiply and Subtract
Multiply [tex]\(x\)[/tex] by the entire denominator [tex]\(3x + 4\)[/tex]:
[tex]\[ x \cdot (3x + 4) = 3x^2 + 4x \][/tex]
Subtract this result from the current numerator:
[tex]\[ (3x^2 - 5x - 12) - (3x^2 + 4x) = -9x - 12 \][/tex]

### Step 6: Third Division
Divide the new leading term of the numerator, [tex]\(-9x\)[/tex], by the leading term of the denominator, [tex]\(3x\)[/tex]:
[tex]\[ \frac{-9x}{3x} = -3 \][/tex]
Write [tex]\(-3\)[/tex] as the next term of the quotient.

### Step 7: Multiply and Subtract
Multiply [tex]\(-3\)[/tex] by the entire denominator [tex]\(3x + 4\)[/tex]:
[tex]\[ -3 \cdot (3x + 4) = -9x - 12 \][/tex]
Subtract this result from the current numerator:
[tex]\[ (-9x - 12) - (-9x - 12) = 0 \][/tex]

### Conclusion
The division process gives us:
[tex]\[ \frac{6x^3 + 11x^2 - 5x - 12}{3x + 4} = 2x^2 + x - 3 \][/tex]
The quotient is [tex]\(2x^2 + x - 3\)[/tex] and the remainder is 0.

So, the simplified version of the polynomial division is indeed:
[tex]\[ 2x^2 + x - 3 \][/tex]
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