Final answer:
To divide the given polynomials, (k³ - 17k + 32) + (k + 5), using synthetic division, rearrange the terms, perform synthetic division step-by-step, and obtain the quotient polynomial as k² - 4k + 15.
Step-by-step explanation:
To divide the polynomials (k³ −17k+32) + (k+5) using synthetic division, we first arrange the terms in descending order of powers. The expression becomes k³ - 17k + k + 32 + 5.
Then, we identify the divisor, which is k+5.
To perform synthetic division, we divide the leading coefficient of the dividend (k³) by the leading coefficient of the divisor (k).
This gives us k² as the first term of the quotient.
Next, we multiply the divisor (k+5) by the first term of the quotient (k²) and subtract this product from the dividend.
Simplifying the resulting expression, we get (k² - 4k + 32).
Now, we bring down the next term, which is -17k, and repeat the process.
By dividing again, the quotient becomes (k² - 4k + 32) - 17, which simplifies to (k² - 4k + 15).
Finally, we bring down the last term, which is 0, and perform the process one last time.
Dividing, we get the final quotient as k² - 4k + 15.