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g(w)=(w+13)^(3)(w+19)^(2) The polynomial function g is defined. When g(w) is divided by (w+16), the remainder is r. What is the value of |r| ?

g(w)=(w+13)3(w+19)2 g(w)=(w+13)^{3}(w+19)^{2} \newlineThe polynomial function g g is defined. When g(w) g(w) is divided by (w+16) (w+16) , the remainder is r r . What is the value of ∣r∣ |r| ?

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Full solution

Q. g(w)=(w+13)3(w+19)2 g(w)=(w+13)^{3}(w+19)^{2} \newlineThe polynomial function g g is defined. When g(w) g(w) is divided by (w+16) (w+16) , the remainder is r r . What is the value of ∣r∣ |r| ?
  1. Use Remainder Theorem: To find the remainder when g(w)g(w) is divided by (w+16)(w+16), we can use the Remainder Theorem. The Remainder Theorem states that if a polynomial f(x)f(x) is divided by (x−c)(x - c), the remainder is f(c)f(c). In this case, we need to evaluate g(−16)g(-16).
  2. Substitute w=−16w = -16: Let's substitute w=−16w = -16 into the function g(w)g(w) to find the remainder rr.\newlineg(w)=(w+13)3(w+19)2g(w) = (w+13)^{3}(w+19)^{2}\newliner=g(−16)=((−16)+13)3((−16)+19)2r = g(-16) = ((-16)+13)^{3}((-16)+19)^{2}
  3. Calculate values: Now we calculate the values inside the parentheses. r=(−3)3(3)2r = (-3)^{3}(3)^{2}
  4. Calculate powers: Next, we calculate the powers. r=(−27)(9)r = (-27)(9)
  5. Multiply numbers: Now we multiply the two numbers to find the remainder rr.\newliner=−27×9r = -27 \times 9\newliner=−243r = -243
  6. Find absolute value: The question asks for the absolute value of rr, which is ∣r∣|r|.\newline∣r∣=∣−243∣|r| = |-243|
  7. Calculate absolute value: Finally, we calculate the absolute value of −243-243.∣r∣=243|r| = 243

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