Question

4
drag each number to the correct location on the table. each number can be used more than once, but not all numbers will be used.
simplify the given polynomial expressions, and determine the degree and number of terms in each expression.
1 0 6 4 2 5 3
degree number of terms
4x + 2x²(3x - 5)
(-3x⁴ + 5x³ - 12) + (7x³ - x⁵ + 6)
(3x² - 3)(3x² + 3)

Explanation:

Step1: Simplify \(4x + 2x^2(3x - 5)\)

First, distribute \(2x^2\) in \(2x^2(3x - 5)\): \(2x^2 \cdot 3x = 6x^3\), \(2x^2 \cdot (-5) = -10x^2\). So the expression becomes \(4x + 6x^3 - 10x^2\), which is \(6x^3 - 10x^2 + 4x\). The degree is the highest power of \(x\), which is \(3\), and the number of terms is \(3\).

Step2: Simplify \((-3x^4 + 5x^3 - 12) + (7x^3 - x^5 + 6)\)

Combine like terms: \(-x^5 - 3x^4 + (5x^3 + 7x^3) + (-12 + 6)= -x^5 - 3x^4 + 12x^3 - 6\). The degree is \(5\) (highest power of \(x\)), and the number of terms is \(4\).

Step3: Simplify \((3x^2 - 3)(3x^2 + 3)\)

Use the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 3x^2\), \(b = 3\). So it becomes \((3x^2)^2 - 3^2 = 9x^4 - 9\). The degree is \(4\), and the number of terms is \(2\).

Answer:

For \(4x + 2x^2(3x - 5)\): Degree = 3, Number of Terms = 3
For \((-3x^4 + 5x^3 - 12) + (7x^3 - x^5 + 6)\): Degree = 5, Number of Terms = 4
For \((3x^2 - 3)(3x^2 + 3)\): Degree = 4, Number of Terms = 2