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Answer :
We are given the polynomial
[tex]$$
x^2 + 5x - 14,
$$[/tex]
and we need to divide it by
[tex]$$
x - 2.
$$[/tex]
Since the divisor is of the form [tex]$x - c$[/tex], we use synthetic division with [tex]$c = 2$[/tex].
Step 1. Set up synthetic division:
List the coefficients of the polynomial:
- Coefficient of [tex]$x^2$[/tex] is [tex]$1$[/tex],
- Coefficient of [tex]$x$[/tex] is [tex]$5$[/tex],
- The constant term is [tex]$-14$[/tex].
So we write:
[tex]$$
1 \quad 5 \quad -14.
$$[/tex]
Write the number [tex]$2$[/tex] (the zero of [tex]$x-2$[/tex]) to the left.
Step 2. Perform synthetic division:
1. Bring down the first coefficient:
Write [tex]$1$[/tex] below the line.
2. Multiply and add:
Multiply the number you brought down by [tex]$2$[/tex]:
[tex]$$1 \times 2 = 2.$$[/tex]
Add this to the second coefficient:
[tex]$$5 + 2 = 7.$$[/tex]
Write [tex]$7$[/tex] in the next position in the result row.
3. Multiply and add again:
Multiply [tex]$7$[/tex] by [tex]$2$[/tex]:
[tex]$$7 \times 2 = 14.$$[/tex]
Add this to the constant term:
[tex]$$-14 + 14 = 0.$$[/tex]
Write [tex]$0$[/tex] as the remainder.
Step 3. Interpret the result:
The numbers below the line (except the remainder) give the coefficients of the quotient polynomial. In this case, we have:
[tex]$$
1 \quad 7
$$[/tex]
This represents the polynomial
[tex]$$
x + 7.
$$[/tex]
The remainder is [tex]$0$[/tex], which confirms that [tex]$x-2$[/tex] is a factor of the polynomial.
Final Answer:
The quotient in polynomial form is
[tex]$$
x + 7.
$$[/tex]
Thus, the correct choice is D.
[tex]$$
x^2 + 5x - 14,
$$[/tex]
and we need to divide it by
[tex]$$
x - 2.
$$[/tex]
Since the divisor is of the form [tex]$x - c$[/tex], we use synthetic division with [tex]$c = 2$[/tex].
Step 1. Set up synthetic division:
List the coefficients of the polynomial:
- Coefficient of [tex]$x^2$[/tex] is [tex]$1$[/tex],
- Coefficient of [tex]$x$[/tex] is [tex]$5$[/tex],
- The constant term is [tex]$-14$[/tex].
So we write:
[tex]$$
1 \quad 5 \quad -14.
$$[/tex]
Write the number [tex]$2$[/tex] (the zero of [tex]$x-2$[/tex]) to the left.
Step 2. Perform synthetic division:
1. Bring down the first coefficient:
Write [tex]$1$[/tex] below the line.
2. Multiply and add:
Multiply the number you brought down by [tex]$2$[/tex]:
[tex]$$1 \times 2 = 2.$$[/tex]
Add this to the second coefficient:
[tex]$$5 + 2 = 7.$$[/tex]
Write [tex]$7$[/tex] in the next position in the result row.
3. Multiply and add again:
Multiply [tex]$7$[/tex] by [tex]$2$[/tex]:
[tex]$$7 \times 2 = 14.$$[/tex]
Add this to the constant term:
[tex]$$-14 + 14 = 0.$$[/tex]
Write [tex]$0$[/tex] as the remainder.
Step 3. Interpret the result:
The numbers below the line (except the remainder) give the coefficients of the quotient polynomial. In this case, we have:
[tex]$$
1 \quad 7
$$[/tex]
This represents the polynomial
[tex]$$
x + 7.
$$[/tex]
The remainder is [tex]$0$[/tex], which confirms that [tex]$x-2$[/tex] is a factor of the polynomial.
Final Answer:
The quotient in polynomial form is
[tex]$$
x + 7.
$$[/tex]
Thus, the correct choice is D.
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