Answer :
To factor the polynomial [tex]\( x^4 - 2x^3 - 19x^2 + 32x + 48 \)[/tex] by [tex]\((x + 1)\)[/tex], you can use synthetic division or polynomial long division. Here’s how you can do it using synthetic division:
1. Set up synthetic division: To divide by [tex]\((x + 1)\)[/tex], you use the root [tex]\(x = -1\)[/tex].
2. Write the coefficients: The coefficients of the polynomial are [tex]\(1, -2, -19, 32, 48\)[/tex].
3. Perform synthetic division:
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by the number just written (1), and write the result below the next coefficient:
[tex]\[
-1 \times 1 = -1
\][/tex]
- Add the result to the next coefficient:
[tex]\[
-2 + (-1) = -3
\][/tex]
- Repeat the multiplication and addition process:
[tex]\[
-1 \times -3 = 3 \quad \Rightarrow \quad -19 + 3 = -16
\][/tex]
[tex]\[
-1 \times -16 = 16 \quad \Rightarrow \quad 32 + 16 = 48
\][/tex]
[tex]\[
-1 \times 48 = -48 \quad \Rightarrow \quad 48 + (-48) = 0
\][/tex]
4. Write the result: The numbers at the bottom are the coefficients of the quotient polynomial:
[tex]\[
1, -3, -16, 48
\][/tex]
This corresponds to the polynomial:
[tex]\[
x^3 - 3x^2 - 16x + 48
\][/tex]
Since the remainder is 0, [tex]\(x + 1\)[/tex] is indeed a factor.
Therefore, the complete factorization of the given polynomial is:
[tex]\[
(x + 1)(x^3 - 3x^2 - 16x + 48)
\][/tex]
Now, you can further factor the cubic polynomial [tex]\(x^3 - 3x^2 - 16x + 48\)[/tex] if possible to find other factors. However, at this step, we have successfully factored out [tex]\((x + 1)\)[/tex].
1. Set up synthetic division: To divide by [tex]\((x + 1)\)[/tex], you use the root [tex]\(x = -1\)[/tex].
2. Write the coefficients: The coefficients of the polynomial are [tex]\(1, -2, -19, 32, 48\)[/tex].
3. Perform synthetic division:
- Bring down the first coefficient: [tex]\(1\)[/tex].
- Multiply [tex]\(-1\)[/tex] by the number just written (1), and write the result below the next coefficient:
[tex]\[
-1 \times 1 = -1
\][/tex]
- Add the result to the next coefficient:
[tex]\[
-2 + (-1) = -3
\][/tex]
- Repeat the multiplication and addition process:
[tex]\[
-1 \times -3 = 3 \quad \Rightarrow \quad -19 + 3 = -16
\][/tex]
[tex]\[
-1 \times -16 = 16 \quad \Rightarrow \quad 32 + 16 = 48
\][/tex]
[tex]\[
-1 \times 48 = -48 \quad \Rightarrow \quad 48 + (-48) = 0
\][/tex]
4. Write the result: The numbers at the bottom are the coefficients of the quotient polynomial:
[tex]\[
1, -3, -16, 48
\][/tex]
This corresponds to the polynomial:
[tex]\[
x^3 - 3x^2 - 16x + 48
\][/tex]
Since the remainder is 0, [tex]\(x + 1\)[/tex] is indeed a factor.
Therefore, the complete factorization of the given polynomial is:
[tex]\[
(x + 1)(x^3 - 3x^2 - 16x + 48)
\][/tex]
Now, you can further factor the cubic polynomial [tex]\(x^3 - 3x^2 - 16x + 48\)[/tex] if possible to find other factors. However, at this step, we have successfully factored out [tex]\((x + 1)\)[/tex].
