Express (x + 6)2 as a trinomial in standard form.
The Correct Answer and Explanation is:
To express (x+6)2(x + 6)^2(x+6)2 as a trinomial in standard form, we start by applying the formula for the square of a binomial:(a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2(a+b)2=a2+2ab+b2
Here, a=xa = xa=x and b=6b = 6b=6. Now, substitute these values into the formula:(x+6)2=x2+2(x)(6)+62(x + 6)^2 = x^2 + 2(x)(6) + 6^2(x+6)2=x2+2(x)(6)+62
First, simplify each term:
- x2x^2×2 remains as is.
- 2(x)(6)=12×2(x)(6) = 12×2(x)(6)=12x.
- 62=366^2 = 3662=36.
Now, put everything together:(x+6)2=x2+12x+36(x + 6)^2 = x^2 + 12x + 36(x+6)2=x2+12x+36
So, the trinomial in standard form is:x2+12x+36x^2 + 12x + 36×2+12x+36
Explanation:
The expression (x+6)2(x + 6)^2(x+6)2 represents a binomial squared, meaning it is the product of the binomial (x+6)(x + 6)(x+6) with itself. To simplify this, we use the square of a binomial formula. This formula breaks the squaring process into three parts:
- Square the first term: The square of xxx is x2x^2×2.
- Multiply the two terms and double it: Multiply xxx and 666, which gives 6x6x6x, and then double it to get 12x12x12x.
- Square the second term: The square of 666 is 363636.
After simplifying all terms, we obtain the trinomial x2+12x+36x^2 + 12x + 36×2+12x+36. This is the expanded form of the binomial expression (x+6)2(x + 6)^2(x+6)2, and it is written in standard form because it is in the form of a polynomial with terms in descending powers of xxx.
