Q:

Match each polynomial function with one of its factors.f(x) = x3 − 3x2 − 13x + 15f(x) = x4 + 3x3 − 8x2 + 5x − 25f(x) = x3 − 2x2 − x + 2f(x) = -x3 + 13x − 12x − 2 -->x + 3 -->x + 4 -->x + 5 -->

Accepted Solution

A:
Answer:[tex]x-2 => f(x)=x^{3}-2x^{2}-x+2\\x+3=>f(x)=x^{3}-3x^{2} -13x+15\\x+4=>f(x)=-x^{3}+13x-12\\x+5=>f(x)=x^{4}+3x^{3}-8x^{2}+5x-25[/tex]Step-by-step explanation:The value of a function will be zero if the factor is put in it. In order to check whether a factor is of a function or not we will put the value of x from that factor in the function:Sox-2 = 0 => x=2Putting in first function[tex]x^{3}-3x^{2} -13x+15\\=(2)^{3}-3(2)^{2} -13(2)+15\\=8-3(4)-26+15\\=8-12-26+15\\=23-38\\=-15 \neq 0\\[/tex]So x-2 is not a factor of first function.Putting in second function[tex]f(x)=x^{4}+3x^{3}-8x^{2}+5x-25\\ =(2)^{4}+3(2)^{3}-8(2)^{2}+5(2)-25\\=16+3(8)-8(4)+10-25\\=16+24-32+10-25\\=-7\neq 0[/tex]So x-2 is also not a factor of second function.Putting in third function:[tex]f(x)=x^{3}-2x^{2}-x+2\\ =(2)^{3}-2(2)^{2}-2+2\\=8-2(4)-2+2\\=8-8-2+2\\=0[/tex]So x-2 is factor of third function. ...........................For x+3x+3=0x=-3First function:[tex]f(x)=x^{3}-3x^{2} -13x+15\\=(-3)^{3}-3(-3)^{2} -13(-3)+15\\=-27-3(9)+39+15\\=-27-27+39+15\\=-54+54\\=0\\[/tex]So x+3 is a factor of first function..............................For x+4x+4=0x=-4As we have already found the factors of first and third function, we will now only check second and fourth function.[tex]f(x)=x^{4}+3x^{3}-8x^{2}+5x-25\\ =(-4)^{4}+3(-4)^{3}-8(-4)^{2}+5(-4)-25\\=256+3(-64)-8(16)-20-25\\=256-192-128-20-25\\-109\neq 0[/tex]So x+4 is not a factor of second function. Putting in fourth function:[tex]f(x)=-x^{3}+13x-12\\ =-(-4)^{3}+13(-4)-12\\=64-52-12\\=64-64\\=0\\[/tex]So x+4 is a factor of fourth function..........................For x+5=0x=-5Since only one function is remaining we'll only check for that.[tex]f(x)=x^{4}+3x^{3}-8x^{2}+5x-25\\ =(-5)^{4}+3(-5)^{3}-8(-5)^{2}+5(-5)-25\\=625+3(-125)-8(25)-25-25\\=625-375-200-25-25\\=0\\[/tex]