Discover the best answers at Westonci.ca, where experts share their insights and knowledge with you. Get detailed and precise answers to your questions from a dedicated community of experts on our Q&A platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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]
Thanks for stopping by. We are committed to providing the best answers for all your questions. See you again soon. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.