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Solve Polynomial Division: (8x³ + 4x² - 2x + 16) ÷ (2x + 3) | Find Quotient and Remainder
Mathematics
College Freshman (MTH099 Developmental Math)
Question Content
Divide: $(8x^3 + 4x^2 - 2x + 16) \\div (2x + 3)$. Your answer should give the quotient and the remainder.
Correct Answer
Quotient: $4x^2 - 4x + 5$, Remainder: $1$
Detailed Solution Steps
1
Step 1: Use polynomial long division. Start by dividing the leading term of the dividend $8x^3$ by the leading term of the divisor $2x$, which gives $4x^2$. This is the first term of the quotient.
2
Step 2: Multiply the entire divisor $(2x+3)$ by $4x^2$: $4x^2(2x+3)=8x^3+12x^2$. Subtract this from the original dividend: $(8x^3 + 4x^2 - 2x + 16) - (8x^3+12x^2) = -8x^2 -2x +16$.
3
Step 3: Divide the leading term of the new polynomial $-8x^2$ by $2x$, which gives $-4x$. Add this to the quotient, so the quotient is now $4x^2-4x$.
4
Step 4: Multiply $(2x+3)$ by $-4x$: $-4x(2x+3)=-8x^2-12x$. Subtract this from $-8x^2 -2x +16$: $(-8x^2 -2x +16) - (-8x^2-12x) = 10x +16$.
5
Step 5: Divide the leading term $10x$ by $2x$, which gives $5$. Add this to the quotient, making the full quotient $4x^2-4x+5$.
6
Step 6: Multiply $(2x+3)$ by $5$: $5(2x+3)=10x+15$. Subtract this from $10x +16$: $(10x +16)-(10x+15)=1$. This is the remainder, which has a lower degree than the divisor, so we stop here.
Knowledge Points Involved
1
Polynomial Long Division
A method for dividing a polynomial by another polynomial of the same or lower degree, similar to integer long division. It involves repeatedly dividing leading terms, multiplying the divisor by the result, subtracting, and repeating until the remaining polynomial has a lower degree than the divisor.
2
Leading Term of a Polynomial
The term in a polynomial with the highest exponent of the variable. It determines the behavior of the polynomial and is used to initiate polynomial division by matching degrees with the divisor's leading term.
3
Quotient and Remainder in Polynomial Division
For polynomial division $\\text{Dividend} \\div \\text{Divisor}$, the result is written as $\\text{Quotient} + \\frac{\\text{Remainder}}{\\text{Divisor}}$, where the remainder must have a degree strictly less than the divisor. If the remainder is 0, the divisor is a factor of the dividend.
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