Video: Finding the Value of β€œπ‘˜β€ Which Makes a Polynomial Divisible by a Given Binomial

Find the value of π‘˜ that makes the expression π‘₯Β² βˆ’ π‘˜π‘₯ + 30 divisible by π‘₯ βˆ’ 5.

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Video Transcript

Find the value of π‘˜ that makes the expression π‘₯ squared minus π‘˜π‘₯ plus 30 divisible by π‘₯ minus five.

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

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

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

This means that π‘₯ minus five is a factor of a quadratic expression π‘₯ squared minus 11 π‘₯ plus 30.

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