Operations with polynomials — Harder example
About this video
The instructor simplifies the expression 7k minus the product of (k + 1) and (2k + 2). He expands the binomials, distributes the negative sign across the whole product, and combines like terms to reach negative 2k squared plus 3k minus 2, then uses the leading term to match the answer choice.
You'll be able to
- multiply two binomials using the distributive property.
- distribute a negative sign across a polynomial.
- combine like terms and write a polynomial in standard form.
Key words
- Binomial
- An algebraic expression with exactly two terms, such as k + 1.
- Distributive property
- Multiplying each term in one group by each term in another group.
- Like terms
- Terms with the same variable raised to the same power, which can be added or subtracted.
- Degree
- The exponent on a variable; the highest exponent in a polynomial gives its degree.
- Standard form
- Writing a polynomial with its highest degree term first.
Transcript
- [Instructor] Which of the following expressions is equivalent to 7k, 7k, minus, minus the product of k plus 1 and 2k plus 2? All right. So this is going to be 7k minus, 7k minus the product of k plus one and 2k plus 2. And, so let's see. To kind of expand it all out, we will want to multiply these two expressions, so let's do that first.
So it's going to be, this is going 7k minus, minus, and then let's just multiply this out. You're gonna have k times 2k, which is 2k squared, k times 2, which is gonna be plus 2k, then 1 times 2k, which is going to give us another 2k, and then 1 times 2, which is going to give us plus 2.
So, and we want to be careful. We want to put a parentheses out front 'cause we're gonna subtract all of this stuff right over here. So this is going to be the same as 7k minus, 7k minus, and then in parentheses, we have 2k squared. We can add 2k plus 2k to get 4k, plus 4k plus 2.
And now we can, one way to think about it is we can distribute this negative sign, or you could even view this as minus 1 times all of this. And so this is going to be the same this 7k minus 2k squared, so minus 2k squared, 2k squared, minus 4k, minus 4k, and then minus 2, minus 2.
And let's see. So if we write the highest degree term first, if we write this term, let me do this in a different color, if we were to write this term first, you get negative 2k squared, that's our highest degree term, and actually immediately when you look at the choices, only one of these start with negative 2k squared.
It has negative 2 as the coefficient on the k squared term. So we already know that that one's going to be the choice, but let's just confirm it. So it's negative 2k squared, and then we could add, we could add 7k and negative 4k. Those two are going to add, 7 minus 4, so I'm thinking it's gonna be three of that, so it's gonna be plus 3k, which is what we see right over there.
And then finally, you hae this minus 2, minus 2, which is exactly what we see in this first choice up here.
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