Video: Finding the Value of “𝑘” Which Makes a Polynomial Divisible by a Given Binomial

Find the value of 𝑘 that makes the expression 𝑥² − 𝑘𝑥 + 30 divisible by 𝑥 − 5.

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Video Transcript

Find the value of 𝑘 that makes the expression 𝑥 squared minus 𝑘𝑥 plus 30 divisible by 𝑥 minus five.

If we let 𝑓 of 𝑥 equal 𝑥 squared minus 𝑘𝑥 plus 30, then if the expression is divisible by 𝑥 minus five, this means that 𝑥 minus five is a factor. If 𝑥 minus 𝑎 is a factor of any equation, then we know the 𝑓 of 𝑎 equals zero. In our case 𝑓 of five, must be equal to zero.

Substituting five into the expression gives us five squared minus 𝑘 multiplied by five plus 30 equals zero. Five squared is equal to 25. And 𝑘 multiplied by five can be written five 𝑘. Grouping the like terms gives us 55 minus five 𝑘 equal zero.

Adding five 𝑘 to both sides to balance the equation gives us five 𝑘 equals 55. Dividing both sides of this equation by five gives us an answer of 𝑘 of 11. Therefore, the value of 𝑘 that makes the expression 𝑥 squared minus 𝑘 𝑥 plus 30 divisible by 𝑥 minus five is 𝑘 equals 11.

This means that 𝑥 minus five is a factor of a quadratic expression 𝑥 squared minus 11 𝑥 plus 30.

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