Solve f(x)=(x-2)^2(x-7)^3

$$f ( x ) = ( x - 2 ) ^ { 2 } ( x - 7 ) ^ { 3 }$$

Evaluate

$$\left(x-2\right)^{2}\left(x-7\right)^{3}$$

Solution Steps

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

$$\left(x^{2}-4x+4\right)\left(x-7\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-7\right)^{3}.

$$\left(x^{2}-4x+4\right)\left(x^{3}-21x^{2}+147x-343\right)$$

Use the distributive property to multiply x^{2}-4x+4 by x^{3}-21x^{2}+147x-343 and combine like terms.

$$x^{5}-25x^{4}+235x^{3}-1015x^{2}+1960x-1372$$

Expand

$$x^{5}-25x^{4}+235x^{3}-1015x^{2}+1960x-1372$$

Solution Steps

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

$$\left(x^{2}-4x+4\right)\left(x-7\right)^{3}$$

Use binomial theorem \left(a-b\right)^{3}=a^{3}-3a^{2}b+3ab^{2}-b^{3} to expand \left(x-7\right)^{3}.

$$\left(x^{2}-4x+4\right)\left(x^{3}-21x^{2}+147x-343\right)$$

Use the distributive property to multiply x^{2}-4x+4 by x^{3}-21x^{2}+147x-343 and combine like terms.

$$x^{5}-25x^{4}+235x^{3}-1015x^{2}+1960x-1372$$