Math / Algebra

Questionarning Goals Long and Synthetic (Polynomial) Division - I can use long division and synthetic division to divide polynomials. long division to divide the following polynomials. $\left(30 x^{3}-65 x^{2}+20 x+30\right) \div(10 x+5)$
2. $\frac{90 x^{3}-15 x^{2}-110 x-40}{10 x+5}$

Studdy Solution

STEP 1

1. We are given two polynomial division problems.
2. We need to use long division to divide the polynomials.

STEP 2

1. Divide the polynomial $30x^3 - 65x^2 + 20x + 30$ by $10x + 5$ using long division.
2. Divide the polynomial $90x^3 - 15x^2 - 110x - 40$ by $10x + 5$ using long division.

STEP 3

Set up the long division for the first polynomial division:
1. Write $30x^3 - 65x^2 + 20x + 30$ under the division symbol.
2. Write $10x + 5$ outside the division symbol.

STEP 4

Divide the first term of the dividend $30x^3$ by the first term of the divisor $10x$ :
$\frac{30x^3}{10x} = 3x^2$
Write $3x^2$ as the first term of the quotient.

STEP 5

Multiply $3x^2$ by the divisor $10x + 5$ and subtract from the dividend:
1. Multiply: $3x^2 \times (10x + 5) = 30x^3 + 15x^2$
2. Subtract: $(30x^3 - 65x^2) - (30x^3 + 15x^2) = -80x^2$
Bring down the next term $20x$ .

STEP 6

Divide the new dividend $-80x^2 + 20x$ by the first term of the divisor $10x$ :
$\frac{-80x^2}{10x} = -8x$
Write $-8x$ as the next term of the quotient.

STEP 7

Multiply $-8x$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $-8x \times (10x + 5) = -80x^2 - 40x$
2. Subtract: $(-80x^2 + 20x) - (-80x^2 - 40x) = 60x$
Bring down the next term $30$ .

STEP 8

Divide the new dividend $60x + 30$ by the first term of the divisor $10x$ :
$\frac{60x}{10x} = 6$
Write $6$ as the next term of the quotient.

STEP 9

Multiply $6$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $6 \times (10x + 5) = 60x + 30$
2. Subtract: $(60x + 30) - (60x + 30) = 0$
There is no remainder.
The quotient for the first division is:
$3x^2 - 8x + 6$

STEP 10

Set up the long division for the second polynomial division:
1. Write $90x^3 - 15x^2 - 110x - 40$ under the division symbol.
2. Write $10x + 5$ outside the division symbol.

STEP 11

Divide the first term of the dividend $90x^3$ by the first term of the divisor $10x$ :
$\frac{90x^3}{10x} = 9x^2$
Write $9x^2$ as the first term of the quotient.

STEP 12

Multiply $9x^2$ by the divisor $10x + 5$ and subtract from the dividend:
1. Multiply: $9x^2 \times (10x + 5) = 90x^3 + 45x^2$
2. Subtract: $(90x^3 - 15x^2) - (90x^3 + 45x^2) = -60x^2$
Bring down the next term $-110x$ .

STEP 13

Divide the new dividend $-60x^2 - 110x$ by the first term of the divisor $10x$ :
$\frac{-60x^2}{10x} = -6x$
Write $-6x$ as the next term of the quotient.

STEP 14

Multiply $-6x$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $-6x \times (10x + 5) = -60x^2 - 30x$
2. Subtract: $(-60x^2 - 110x) - (-60x^2 - 30x) = -80x$
Bring down the next term $-40$ .

STEP 15

Divide the new dividend $-80x - 40$ by the first term of the divisor $10x$ :
$\frac{-80x}{10x} = -8$
Write $-8$ as the next term of the quotient.

STEP 16

Multiply $-8$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $-8 \times (10x + 5) = -80x - 40$
2. Subtract: $(-80x - 40) - (-80x - 40) = 0$
There is no remainder.
The quotient for the second division is:
$9x^2 - 6x - 8$

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Questionarning Goals Long and Synthetic (Polynomial) Division - I can use long division and synthetic division to divide polynomials. long division to divide the following polynomials. $\left(30 x^{3}-65 x^{2}+20 x+30\right) \div(10 x+5)$
2. $\frac{90 x^{3}-15 x^{2}-110 x-40}{10 x+5}$

Studdy Solution

STEP 1

1. We are given two polynomial division problems.
2. We need to use long division to divide the polynomials.

STEP 2

1. Divide the polynomial $30x^3 - 65x^2 + 20x + 30$ by $10x + 5$ using long division.
2. Divide the polynomial $90x^3 - 15x^2 - 110x - 40$ by $10x + 5$ using long division.

STEP 3

Set up the long division for the first polynomial division:
1. Write $30x^3 - 65x^2 + 20x + 30$ under the division symbol.
2. Write $10x + 5$ outside the division symbol.

STEP 4

Divide the first term of the dividend $30x^3$ by the first term of the divisor $10x$ :
$\frac{30x^3}{10x} = 3x^2$
Write $3x^2$ as the first term of the quotient.

STEP 5

Multiply $3x^2$ by the divisor $10x + 5$ and subtract from the dividend:
1. Multiply: $3x^2 \times (10x + 5) = 30x^3 + 15x^2$
2. Subtract: $(30x^3 - 65x^2) - (30x^3 + 15x^2) = -80x^2$
Bring down the next term $20x$ .

STEP 6

Divide the new dividend $-80x^2 + 20x$ by the first term of the divisor $10x$ :
$\frac{-80x^2}{10x} = -8x$
Write $-8x$ as the next term of the quotient.

STEP 7

Multiply $-8x$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $-8x \times (10x + 5) = -80x^2 - 40x$
2. Subtract: $(-80x^2 + 20x) - (-80x^2 - 40x) = 60x$
Bring down the next term $30$ .

STEP 8

Divide the new dividend $60x + 30$ by the first term of the divisor $10x$ :
$\frac{60x}{10x} = 6$
Write $6$ as the next term of the quotient.

STEP 9

Multiply $6$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $6 \times (10x + 5) = 60x + 30$
2. Subtract: $(60x + 30) - (60x + 30) = 0$
There is no remainder.
The quotient for the first division is:
$3x^2 - 8x + 6$

STEP 10

Set up the long division for the second polynomial division:
1. Write $90x^3 - 15x^2 - 110x - 40$ under the division symbol.
2. Write $10x + 5$ outside the division symbol.

STEP 11

Divide the first term of the dividend $90x^3$ by the first term of the divisor $10x$ :
$\frac{90x^3}{10x} = 9x^2$
Write $9x^2$ as the first term of the quotient.

STEP 12

Multiply $9x^2$ by the divisor $10x + 5$ and subtract from the dividend:
1. Multiply: $9x^2 \times (10x + 5) = 90x^3 + 45x^2$
2. Subtract: $(90x^3 - 15x^2) - (90x^3 + 45x^2) = -60x^2$
Bring down the next term $-110x$ .

STEP 13

Divide the new dividend $-60x^2 - 110x$ by the first term of the divisor $10x$ :
$\frac{-60x^2}{10x} = -6x$
Write $-6x$ as the next term of the quotient.

STEP 14

Multiply $-6x$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $-6x \times (10x + 5) = -60x^2 - 30x$
2. Subtract: $(-60x^2 - 110x) - (-60x^2 - 30x) = -80x$
Bring down the next term $-40$ .

STEP 15

Divide the new dividend $-80x - 40$ by the first term of the divisor $10x$ :
$\frac{-80x}{10x} = -8$
Write $-8$ as the next term of the quotient.

STEP 16

Multiply $-8$ by the divisor $10x + 5$ and subtract from the new dividend:
1. Multiply: $-8 \times (10x + 5) = -80x - 40$
2. Subtract: $(-80x - 40) - (-80x - 40) = 0$
There is no remainder.
The quotient for the second division is:
$9x^2 - 6x - 8$

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Studdy Solution

STEP 1

1. We are given two polynomial division problems.2. We need to use long division to divide the polynomials.

STEP 2

1. Divide the polynomial 30x3−65x2+20x+3030x^3 - 65x^2 + 20x + 3030x3−65x2+20x+30 by 10x+510x + 510x+5 using long division.2. Divide the polynomial 90x3−15x2−110x−4090x^3 - 15x^2 - 110x - 4090x3−15x2−110x−40 by 10x+510x + 510x+5 using long division.

STEP 3

Set up the long division for the first polynomial division:1. Write 30x3−65x2+20x+3030x^3 - 65x^2 + 20x + 3030x3−65x2+20x+30 under the division symbol.2. Write 10x+510x + 510x+5 outside the division symbol.

STEP 4

Divide the first term of the dividend 30x330x^330x3 by the first term of the divisor 10x10x10x:30x310x=3x2 \frac{30x^3}{10x} = 3x^2 10x30x3​=3x2Write 3x23x^23x2 as the first term of the quotient.

STEP 5

STEP 6

Divide the new dividend −80x2+20x-80x^2 + 20x−80x2+20x by the first term of the divisor 10x10x10x:−80x210x=−8x \frac{-80x^2}{10x} = -8x 10x−80x2​=−8xWrite −8x-8x−8x as the next term of the quotient.

STEP 7

STEP 8

Divide the new dividend 60x+3060x + 3060x+30 by the first term of the divisor 10x10x10x:60x10x=6 \frac{60x}{10x} = 6 10x60x​=6Write 666 as the next term of the quotient.

STEP 9

STEP 10

Set up the long division for the second polynomial division:1. Write 90x3−15x2−110x−4090x^3 - 15x^2 - 110x - 4090x3−15x2−110x−40 under the division symbol.2. Write 10x+510x + 510x+5 outside the division symbol.

STEP 11

Divide the first term of the dividend 90x390x^390x3 by the first term of the divisor 10x10x10x:90x310x=9x2 \frac{90x^3}{10x} = 9x^2 10x90x3​=9x2Write 9x29x^29x2 as the first term of the quotient.

STEP 12

STEP 13

Divide the new dividend −60x2−110x-60x^2 - 110x−60x2−110x by the first term of the divisor 10x10x10x:−60x210x=−6x \frac{-60x^2}{10x} = -6x 10x−60x2​=−6xWrite −6x-6x−6x as the next term of the quotient.

STEP 14

STEP 15

Divide the new dividend −80x−40-80x - 40−80x−40 by the first term of the divisor 10x10x10x:−80x10x=−8 \frac{-80x}{10x} = -8 10x−80x​=−8Write −8-8−8 as the next term of the quotient.

STEP 16

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1. We are given two polynomial division problems.
2. We need to use long division to divide the polynomials.

1. Divide the polynomial $30x^3 - 65x^2 + 20x + 30$ by $10x + 5$ using long division.
2. Divide the polynomial $90x^3 - 15x^2 - 110x - 40$ by $10x + 5$ using long division.

Set up the long division for the first polynomial division:
1. Write $30x^3 - 65x^2 + 20x + 30$ under the division symbol.
2. Write $10x + 5$ outside the division symbol.

Divide the first term of the dividend $30x^3$ by the first term of the divisor $10x$ :
$\frac{30x^3}{10x} = 3x^2$
Write $3x^2$ as the first term of the quotient.

Divide the new dividend $-80x^2 + 20x$ by the first term of the divisor $10x$ :
$\frac{-80x^2}{10x} = -8x$
Write $-8x$ as the next term of the quotient.

Divide the new dividend $60x + 30$ by the first term of the divisor $10x$ :
$\frac{60x}{10x} = 6$
Write $6$ as the next term of the quotient.

Set up the long division for the second polynomial division:
1. Write $90x^3 - 15x^2 - 110x - 40$ under the division symbol.
2. Write $10x + 5$ outside the division symbol.

Divide the first term of the dividend $90x^3$ by the first term of the divisor $10x$ :
$\frac{90x^3}{10x} = 9x^2$
Write $9x^2$ as the first term of the quotient.

Divide the new dividend $-60x^2 - 110x$ by the first term of the divisor $10x$ :
$\frac{-60x^2}{10x} = -6x$
Write $-6x$ as the next term of the quotient.

Divide the new dividend $-80x - 40$ by the first term of the divisor $10x$ :
$\frac{-80x}{10x} = -8$
Write $-8$ as the next term of the quotient.