Welcome to Westonci.ca, the place where your questions find answers from a community of knowledgeable experts. Discover precise answers to your questions from a wide range of experts on our user-friendly Q&A platform. Connect with a community of professionals ready to provide precise solutions to your questions quickly and accurately.
Sagot :
To find [tex]\( p(-3) \)[/tex] using synthetic substitution for the polynomial [tex]\( p(x) = x^3 + 6x^2 + 7x - 25 \)[/tex], we will follow these steps:
1. Identify the coefficients of the polynomial:
The polynomial is [tex]\( p(x) = x^3 + 6x^2 + 7x - 25 \)[/tex].
The coefficients are: 1 (for [tex]\( x^3 \)[/tex]), 6 (for [tex]\( x^2 \)[/tex]), 7 (for [tex]\( x \)[/tex]), and -25 (constant term).
2. Set up the synthetic division:
We are evaluating [tex]\( p(x) \)[/tex] at [tex]\( x = -3 \)[/tex]. Arrange the coefficients in a row:
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ \end{array} \][/tex]
3. Perform the synthetic substitution step-by-step:
- Step 1: Bring down the first coefficient (1) to the bottom row:
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & & & \\ & 1 & & & \\ \end{array} \][/tex]
- Step 2: Multiply the first coefficient by [tex]\( -3 \)[/tex] and write the result under the second coefficient:
[tex]\[ 1 \times (-3) = -3 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& & \\ & 1 & & & \\ \end{array} \][/tex]
- Step 3: Add the second coefficient (6) and the result from the multiplication (-3):
[tex]\[ 6 + (-3) = 3 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& & \\ & 1 & 3 & & \\ \end{array} \][/tex]
- Step 4: Multiply the result (3) by [tex]\( -3 \)[/tex] and write it under the third coefficient:
[tex]\[ 3 \times (-3) = -9 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& \\ & 1 & 3 & & \\ \end{array} \][/tex]
- Step 5: Add the third coefficient (7) and the result from the multiplication (-9):
[tex]\[ 7 + (-9) = -2 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& \\ & 1 & 3 & -2& \\ \end{array} \][/tex]
- Step 6: Multiply the result (-2) by [tex]\( -3 \)[/tex] and write it under the fourth coefficient:
[tex]\[ -2 \times (-3) = 6 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& 6 \\ & 1 & 3 & -2 & \\ \end{array} \][/tex]
- Step 7: Add the fourth coefficient (-25) and the result from the multiplication (6):
[tex]\[ -25 + 6 = -19 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& 6 \\ & 1 & 3 & -2 & -19 \\ \end{array} \][/tex]
So, we see that the result of [tex]\( p(-3) \)[/tex] is -19. Thus:
[tex]\[ p(-3) = -19 \][/tex]
1. Identify the coefficients of the polynomial:
The polynomial is [tex]\( p(x) = x^3 + 6x^2 + 7x - 25 \)[/tex].
The coefficients are: 1 (for [tex]\( x^3 \)[/tex]), 6 (for [tex]\( x^2 \)[/tex]), 7 (for [tex]\( x \)[/tex]), and -25 (constant term).
2. Set up the synthetic division:
We are evaluating [tex]\( p(x) \)[/tex] at [tex]\( x = -3 \)[/tex]. Arrange the coefficients in a row:
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ \end{array} \][/tex]
3. Perform the synthetic substitution step-by-step:
- Step 1: Bring down the first coefficient (1) to the bottom row:
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & & & \\ & 1 & & & \\ \end{array} \][/tex]
- Step 2: Multiply the first coefficient by [tex]\( -3 \)[/tex] and write the result under the second coefficient:
[tex]\[ 1 \times (-3) = -3 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& & \\ & 1 & & & \\ \end{array} \][/tex]
- Step 3: Add the second coefficient (6) and the result from the multiplication (-3):
[tex]\[ 6 + (-3) = 3 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& & \\ & 1 & 3 & & \\ \end{array} \][/tex]
- Step 4: Multiply the result (3) by [tex]\( -3 \)[/tex] and write it under the third coefficient:
[tex]\[ 3 \times (-3) = -9 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& \\ & 1 & 3 & & \\ \end{array} \][/tex]
- Step 5: Add the third coefficient (7) and the result from the multiplication (-9):
[tex]\[ 7 + (-9) = -2 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& \\ & 1 & 3 & -2& \\ \end{array} \][/tex]
- Step 6: Multiply the result (-2) by [tex]\( -3 \)[/tex] and write it under the fourth coefficient:
[tex]\[ -2 \times (-3) = 6 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& 6 \\ & 1 & 3 & -2 & \\ \end{array} \][/tex]
- Step 7: Add the fourth coefficient (-25) and the result from the multiplication (6):
[tex]\[ -25 + 6 = -19 \][/tex]
[tex]\[ \begin{array}{r|rrrr} -3 & 1 & 6 & 7 & -25 \\ & & -3& -9& 6 \\ & 1 & 3 & -2 & -19 \\ \end{array} \][/tex]
So, we see that the result of [tex]\( p(-3) \)[/tex] is -19. Thus:
[tex]\[ p(-3) = -19 \][/tex]
We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. We appreciate your time. Please come back anytime for the latest information and answers to your questions. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.