Solve (x-13)((x-7)(x-7)+32)+5(-2(x-7)+40)+2(8+5(x-7))

$$( x - 13 ) ( ( x - 7 ) ( x - 7 ) + 32 ) + 5 ( - 2 ( x - 7 ) + 40 ) + 2 ( 8 + 5 ( x - 7 ) )$$

Evaluate

$$x^{3}-27x^{2}+263x-837$$

Solution Steps

Multiply x-7 and x-7 to get \left(x-7\right)^{2}.

$$\left(x-13\right)\left(\left(x-7\right)^{2}+32\right)+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-7\right)^{2}.

$$\left(x-13\right)\left(x^{2}-14x+49+32\right)+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Add 49 and 32 to get 81.

$$\left(x-13\right)\left(x^{2}-14x+81\right)+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Apply the distributive property by multiplying each term of x-13 by each term of x^{2}-14x+81.

$$x^{3}-14x^{2}+81x-13x^{2}+182x-1053+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Combine -14x^{2} and -13x^{2} to get -27x^{2}.

$$x^{3}-27x^{2}+81x+182x-1053+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Combine 81x and 182x to get 263x.

$$x^{3}-27x^{2}+263x-1053+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Use the distributive property to multiply -2 by x-7.

$$x^{3}-27x^{2}+263x-1053+5\left(-2x+14+40\right)+2\left(8+5\left(x-7\right)\right)$$

Add 14 and 40 to get 54.

$$x^{3}-27x^{2}+263x-1053+5\left(-2x+54\right)+2\left(8+5\left(x-7\right)\right)$$

Use the distributive property to multiply 5 by -2x+54.

$$x^{3}-27x^{2}+263x-1053-10x+270+2\left(8+5\left(x-7\right)\right)$$

Combine 263x and -10x to get 253x.

$$x^{3}-27x^{2}+253x-1053+270+2\left(8+5\left(x-7\right)\right)$$

Add -1053 and 270 to get -783.

$$x^{3}-27x^{2}+253x-783+2\left(8+5\left(x-7\right)\right)$$

Use the distributive property to multiply 5 by x-7.

$$x^{3}-27x^{2}+253x-783+2\left(8+5x-35\right)$$

Subtract 35 from 8 to get -27.

$$x^{3}-27x^{2}+253x-783+2\left(-27+5x\right)$$

Use the distributive property to multiply 2 by -27+5x.

$$x^{3}-27x^{2}+253x-783-54+10x$$

Subtract 54 from -783 to get -837.

$$x^{3}-27x^{2}+253x-837+10x$$

Combine 253x and 10x to get 263x.

$$x^{3}-27x^{2}+263x-837$$

Expand

$$x^{3}-27x^{2}+263x-837$$

Solution Steps

Multiply x-7 and x-7 to get \left(x-7\right)^{2}.

$$\left(x-13\right)\left(\left(x-7\right)^{2}+32\right)+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x-7\right)^{2}.

$$\left(x-13\right)\left(x^{2}-14x+49+32\right)+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Add 49 and 32 to get 81.

$$\left(x-13\right)\left(x^{2}-14x+81\right)+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Apply the distributive property by multiplying each term of x-13 by each term of x^{2}-14x+81.

$$x^{3}-14x^{2}+81x-13x^{2}+182x-1053+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Combine -14x^{2} and -13x^{2} to get -27x^{2}.

$$x^{3}-27x^{2}+81x+182x-1053+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Combine 81x and 182x to get 263x.

$$x^{3}-27x^{2}+263x-1053+5\left(-2\left(x-7\right)+40\right)+2\left(8+5\left(x-7\right)\right)$$

Use the distributive property to multiply -2 by x-7.

$$x^{3}-27x^{2}+263x-1053+5\left(-2x+14+40\right)+2\left(8+5\left(x-7\right)\right)$$

Add 14 and 40 to get 54.

$$x^{3}-27x^{2}+263x-1053+5\left(-2x+54\right)+2\left(8+5\left(x-7\right)\right)$$

Use the distributive property to multiply 5 by -2x+54.

$$x^{3}-27x^{2}+263x-1053-10x+270+2\left(8+5\left(x-7\right)\right)$$

Combine 263x and -10x to get 253x.

$$x^{3}-27x^{2}+253x-1053+270+2\left(8+5\left(x-7\right)\right)$$

Add -1053 and 270 to get -783.

$$x^{3}-27x^{2}+253x-783+2\left(8+5\left(x-7\right)\right)$$

Use the distributive property to multiply 5 by x-7.

$$x^{3}-27x^{2}+253x-783+2\left(8+5x-35\right)$$

Subtract 35 from 8 to get -27.

$$x^{3}-27x^{2}+253x-783+2\left(-27+5x\right)$$

Use the distributive property to multiply 2 by -27+5x.

$$x^{3}-27x^{2}+253x-783-54+10x$$

Subtract 54 from -783 to get -837.

$$x^{3}-27x^{2}+253x-837+10x$$

Combine 253x and 10x to get 263x.

$$x^{3}-27x^{2}+263x-837$$