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avatar+955 

What is the coefficient of $x$ in $(x^3 + x^2 + x + 1)(x^4 - 8x^3 + 17x^2 - 23x + 14)$?

 Dec 8, 2023
 #1
avatar+222 
0

We can expand the polynomial to find the coefficient of x.

 

Step 1: Use the distributive property to multiply the two polynomials:

 

$$ \begin{array}{c r@{}l} &x^3 + x^2 + x + 1 & (x^4 - 8x^3 + 17x^2 - 23x + 14) \ \times & x^4 & x^7 - 8x^6 + 17x^5 - 23x^4 + 14x^3 + x^2 - 8x + 4x - 14 \ & +x^2 & x^6 - 8x^5 + 17x^4 - 23x^3 + 14x^2 + x - 8x + 4x - 14 \ & +x & x^5 - 8x^4 + 17x^3 - 23x^2 + 14x + x - 8x + 4x - 14 \ & + 1 & x^4 - 8x^3 + 17x^2 - 23x + 14 + x - 8x + 4x - 14 \ \end{array} $$

 

Step 2: Combine like terms:

 

$$ \begin{array}{r r c r@{}l} &x^7 & - 3x^6 & +14x^5 & +18x^4 & - 5x^3 & +3x^2 & -14x & -18 &\ & & & & +14x^3 &+4x^2 & -8x& +4x & -14 \ & & & & & & +7x^2 & -14x & -14 \ \cline{2-10} & x^7 & - 3x^6 &+14x^5 & +18x^4 & - 5x^3 &+3x^2 & -14x & -18 & \end{array} $$

 

Therefore, the coefficient of x is −14​.

 Dec 8, 2023
 #2
avatar+67 
+1

WE GOT X WHEN X MULTIPLY BY CONSTANT OR CONSTANT MULTIPLY BY X TERMS . 

HERE ,WHEN X IS MULTIPLY BY 14 THEN WE GOT 14X.

ALSO -23X IS MULTIPLY BY 1 WE GOT -23X.

-23X+14X=-7X.

SO , OUR ANSWER IS -7.

 Dec 9, 2023

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