Question

1.?
what do you need to multiply the divisor by to get a coefficient of -1
how many terms are in the dividend? 4
enter the quotient and remainder.
synthetic division table

Explanation:

Step1: Analyze the Synthetic Division

In synthetic division, the number of terms in the quotient is one less than the number of terms in the dividend. The problem states the dividend has 4 terms. So the quotient should have \( 4 - 1 = 3 \) terms? Wait, no, looking at the synthetic division table, let's check the coefficients. Wait, maybe the dividend is a polynomial, and we are doing synthetic division. Wait, the synthetic division table has entries: let's see the numbers. Wait, maybe the dividend is \( 5x^3 - 37x^2 + 10x + 28 \) (assuming the coefficients from the table: 5, -37, 10, 28) and the divisor is related to the root. Wait, but the question about the quotient and remainder. Wait, the remainder is given as 0? Wait, no, the user's image shows a quotient box and remainder box with remainder 0? Wait, maybe the division is exact. Wait, let's re - evaluate. Wait, the synthetic division process: when we do synthetic division, the last number is the remainder. If the remainder is 0, then the divisor is a factor. Now, the number of terms in the dividend: a polynomial with degree \( n \) has \( n + 1 \) terms. If the dividend has 4 terms, its degree is 3. Then the quotient will have degree \( 3 - 1=2 \), so 3 terms. But looking at the numbers on the left (5, -2, -4) and the synthetic division table (7, 5, -37, 10, 28; and 35, -14, -28). Wait, maybe the divisor is \( x - 7 \)? Wait, no, maybe the first part: "How many terms are in the dividend? 4" So the dividend is a 4 - term polynomial (degree 3). Then when we do synthetic division, the quotient will be a 3 - term polynomial (degree 2) and remainder. But the remainder is 0 here? Wait, maybe the quotient is \( 5x^2 - 2x - 4 \) (from the left - hand numbers: 5, -2, -4) and remainder 0. Let's check: \( (x - 7)(5x^2 - 2x - 4)=5x^3-2x^2 - 4x-35x^2 + 14x + 28=5x^3-37x^2 + 10x + 28 \), which matches the coefficients in the synthetic division table (5, -37, 10, 28). So the quotient is \( 5x^2 - 2x - 4 \) and remainder 0.

Step1: Determine the number of terms in dividend

The problem states the dividend has 4 terms. A polynomial with \( n \) terms has a degree of \( n - 1 \). So a 4 - term polynomial has degree 3.

Step2: Find the degree of the quotient

When dividing a degree - 3 polynomial (dividend) by a degree - 1 polynomial (divisor, since synthetic division is for linear divisors), the quotient will have degree \( 3-1 = 2 \). A degree - 2 polynomial has \( 2 + 1=3 \) terms.

Step3: Identify the quotient from the synthetic division

Looking at the synthetic division table and the numbers on the left (5, -2, -4), we can form the quotient polynomial. The coefficients of the quotient are 5, -2, -4, so the quotient is \( 5x^2-2x - 4 \) and the remainder is 0.

Answer:

Quotient: \( 5x^2 - 2x - 4 \), Remainder: \( 0 \)