Find the product of 1.5x and (2.7-x)
The correct answer and explanation is:
Answer:
To find the product of 1.5×1.5x and (2.7−x)(2.7 – x), we use the distributive property: 1.5x⋅(2.7−x)=1.5x⋅2.7−1.5x⋅x1.5x \cdot (2.7 – x) = 1.5x \cdot 2.7 – 1.5x \cdot x
Now calculate each term:
- 1.5x⋅2.7=4.05×1.5x \cdot 2.7 = 4.05x
- 1.5x⋅x=1.5×21.5x \cdot x = 1.5x^2
So, the product is: 1.5x(2.7−x)=4.05x−1.5×21.5x(2.7 – x) = 4.05x – 1.5x^2
Explanation (Approx. 300 Words):
When multiplying algebraic expressions, especially a monomial (like 1.5×1.5x) with a binomial (like 2.7−x2.7 – x), we use the distributive property. This property allows you to “distribute” the monomial across the terms inside the parentheses.
In this case, the binomial has two terms: 2.7 and -x. The monomial outside the parenthesis is 1.5×1.5x. To multiply 1.5×1.5x by the binomial, you multiply it separately with each term inside the parentheses:
- First Term: Multiply 1.5×1.5x by 2.7: 1.5x⋅2.7=(1.5⋅2.7)x=4.05×1.5x \cdot 2.7 = (1.5 \cdot 2.7)x = 4.05x Here, we multiply the numerical coefficients (1.5 and 2.7), and keep the variable xx attached.
- Second Term: Multiply 1.5×1.5x by −x-x: 1.5x⋅(−x)=−1.5×21.5x \cdot (-x) = -1.5x^2 This time, you’re multiplying the coefficients (1.5 and -1), and the variables x⋅x=x2x \cdot x = x^2.
Now combine the two results: 4.05x−1.5×24.05x – 1.5x^2
This is the final expression for the product.
This expression is in standard polynomial form, where terms are ordered by decreasing exponents. You can also write it as: −1.5×2+4.05x-1.5x^2 + 4.05x
Both forms are mathematically equivalent. The expression represents a quadratic polynomial, and it’s useful in many applications like optimization problems, graphing parabolas, and modeling real-world situations.