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Answer :
To find the quotient when dividing
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3,
$$[/tex]
we first note that the degree of the dividend is 4 and the degree of the divisor is 3. This means the quotient should be a linear polynomial of the form
[tex]$$
Ax+B.
$$[/tex]
Since
[tex]$$
(Ax+B)(x^3-3)=Ax^4+Bx^3-3Ax-3B,
$$[/tex]
we want this product to equal the dividend:
[tex]$$
x^4+5x^3-3x-15.
$$[/tex]
Now we compare the coefficients of like powers of [tex]$x$[/tex] from both sides.
1. Coefficient of [tex]$x^4$[/tex]:
From the product: [tex]$Ax^4$[/tex]
From the dividend: [tex]$1x^4$[/tex]
Thus, we have:
[tex]$$
A=1.
$$[/tex]
2. Coefficient of [tex]$x^3$[/tex]:
From the product: [tex]$Bx^3$[/tex]
From the dividend: [tex]$5x^3$[/tex]
Thus, we have:
[tex]$$
B=5.
$$[/tex]
3. Coefficient of [tex]$x$[/tex]:
From the product (coefficient of [tex]$x$[/tex]): [tex]$-3A$[/tex]
With [tex]$A=1$[/tex], we get:
[tex]$$
-3A = -3,
$$[/tex]
which matches the dividend's coefficient (also [tex]$-3$[/tex]).
4. Constant term:
From the product (constant term): [tex]$-3B$[/tex]
With [tex]$B=5$[/tex], we get:
[tex]$$
-3B = -15,
$$[/tex]
which matches the dividend's constant term.
Since all coefficients match perfectly, the division has no remainder and the quotient is exactly:
[tex]$$
x+5.
$$[/tex]
Thus, the quotient is [tex]$\boxed{x+5}$[/tex].
[tex]$$
x^4+5x^3-3x-15
$$[/tex]
by
[tex]$$
x^3-3,
$$[/tex]
we first note that the degree of the dividend is 4 and the degree of the divisor is 3. This means the quotient should be a linear polynomial of the form
[tex]$$
Ax+B.
$$[/tex]
Since
[tex]$$
(Ax+B)(x^3-3)=Ax^4+Bx^3-3Ax-3B,
$$[/tex]
we want this product to equal the dividend:
[tex]$$
x^4+5x^3-3x-15.
$$[/tex]
Now we compare the coefficients of like powers of [tex]$x$[/tex] from both sides.
1. Coefficient of [tex]$x^4$[/tex]:
From the product: [tex]$Ax^4$[/tex]
From the dividend: [tex]$1x^4$[/tex]
Thus, we have:
[tex]$$
A=1.
$$[/tex]
2. Coefficient of [tex]$x^3$[/tex]:
From the product: [tex]$Bx^3$[/tex]
From the dividend: [tex]$5x^3$[/tex]
Thus, we have:
[tex]$$
B=5.
$$[/tex]
3. Coefficient of [tex]$x$[/tex]:
From the product (coefficient of [tex]$x$[/tex]): [tex]$-3A$[/tex]
With [tex]$A=1$[/tex], we get:
[tex]$$
-3A = -3,
$$[/tex]
which matches the dividend's coefficient (also [tex]$-3$[/tex]).
4. Constant term:
From the product (constant term): [tex]$-3B$[/tex]
With [tex]$B=5$[/tex], we get:
[tex]$$
-3B = -15,
$$[/tex]
which matches the dividend's constant term.
Since all coefficients match perfectly, the division has no remainder and the quotient is exactly:
[tex]$$
x+5.
$$[/tex]
Thus, the quotient is [tex]$\boxed{x+5}$[/tex].
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