Constructing a polynomial that has a certain factor | Algebra II | Khan Academy

TL;DR

By setting the value of p(5) to zero, we can solve for the value of c in the polynomial expression.

Transcript

• So for what value of c, or values of c, is x minus five? Let me write it this way. For what c is x minus five a factor of p of x? I encourage you to pause the video and try to work through it. That p is actually a lower case p. P of x. All right, so I'm assuming you have attempted this. We just have to realize, okay, if x minus five is a factor o... Read More

Key Insights

• ☺️ If x - 5 is a factor of p(x), then p(5) will be equal to zero.
• 😫 Setting p(5) equal to zero, we can solve for the value of c.
• 🫚 A root of a polynomial means that the polynomial is equal to zero at that particular value.

Questions & Answers

Q: How do we determine if x - 5 is a factor of p(x)?

We can write p(x) as (x - 5) multiplied by another polynomial, g(x). If x - 5 is a factor, then p(5) will be equal to zero.

Q: What does it mean for a value to be a root of a polynomial?

A root of a polynomial is a value that makes the polynomial equal to zero. In this case, if x = 5, p(x) will be equal to zero, indicating that 5 is a root of the polynomial.

Q: How do we find the value of c?

By setting p(5) equal to zero, we can substitute the value of 5 into the polynomial expression and solve for c.

Q: What is the final value of c?

After solving the equation p(5) = 185 + 5c = 0, we find that c is equal to -37.

Summary & Key Takeaways

• To find the value of c, we need to determine when x - 5 is a factor of p(x).

• This means that p(x) can be written as (x - 5) multiplied by another polynomial, g(x).

• To find the value of c, we set p(5) equal to zero and solve for c.