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Solve $(5x^2 + 29x - 38) \\div (x + 7)$: Find Quotient and Remainder
Mathematics
Grade 10 (Junior High School)
Question Content
Divide $(5x^2 + 29x - 38) \\div (x + 7)$. Your answer should give the quotient and the remainder.
Correct Answer
Quotient: $5x - 6$, Remainder: 4
Detailed Solution Steps
1
Step 1: Use polynomial long division. First, divide the leading term of the dividend $5x^2$ by the leading term of the divisor $x$, which gives $5x$. This is the first term of the quotient.
2
Step 2: Multiply the entire divisor $(x + 7)$ by $5x$, which results in $5x^2 + 35x$.
3
Step 3: Subtract this product from the original dividend: $(5x^2 + 29x - 38) - (5x^2 + 35x) = -6x - 38$.
4
Step 4: Now divide the leading term of the new polynomial $-6x$ by the leading term of the divisor $x$, which gives $-6$. Add this to the quotient, so the quotient is now $5x - 6$.
5
Step 5: Multiply the divisor $(x + 7)$ by $-6$, which gives $-6x - 42$.
6
Step 6: Subtract this from the polynomial from Step 3: $(-6x - 38) - (-6x - 42) = (-6x - 38 + 6x + 42) = 4$. 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 used to divide a polynomial by another polynomial of lower or equal degree. The process involves dividing leading terms, multiplying the divisor by the resulting term, subtracting the product from the dividend, and repeating until the remaining polynomial (remainder) has a lower degree than the divisor. It is analogous to long division of whole numbers but for algebraic expressions.
2
Leading Term of a Polynomial
The term in a polynomial with the highest power of the variable. For example, in $5x^2 + 29x - 38$, the leading term is $5x^2$. It is used to determine the first term of the quotient in polynomial division.
3
Quotient and Remainder in Polynomial Division
When dividing a polynomial $P(x)$ (dividend) by a polynomial $D(x)$ (divisor), we can write $P(x) = D(x)Q(x) + R(x)$, where $Q(x)$ is the quotient polynomial and $R(x)$ is the remainder. The degree of $R(x)$ must be less than the degree of $D(x)$, or $R(x) = 0$ if the division is exact.
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