1.4 Polynomials

For the following polynomials, identify the degree, the leading term, and the leading coefficient.

1. ⓐ $3 + 2 x^{2} - 4 x^{3}$
2. ⓑ $5 t^{5} - 2 t^{3} + 7 t$
3. ⓒ $6 p - p^{3} - 2$

Identify the degree, leading term, and leading coefficient of the polynomial $4 x^{2} - x^{6} + 2 x - 6.$

Find the sum.

$\left(12 x^{2} + 9 x - 21\right) + \left(4 x^{3} + 8 x^{2} - 5 x + 20\right)$

Find the sum.

$\left(2 x^{3} + 5 x^{2} - x + 1\right) + \left(2 x^{2} - 3 x - 4\right)$

Find the difference.

$\left(7 x^{4} - x^{2} + 6 x + 1\right) - \left(5 x^{3} - 2 x^{2} + 3 x + 2\right)$

Find the difference.

$\left(−7 x^{3} - 7 x^{2} + 6 x - 2\right) - \left(4 x^{3} - 6 x^{2} - x + 7\right)$

Find the product.

$\left(2 x + 1\right) \left(3 x^{2} - x + 4\right)$

Find the product.

$\left(3 x + 2\right) \left(x^{3} - 4 x^{2} + 7\right)$

Use FOIL to find the product.

$\left(x + 7\right) \left(3 x - 5\right)$

Expand $\left(3 x - 8\right)^{2} .$

Expand $\left(4 x - 1\right)^{2} .$

Multiply $\left(9 x + 4\right) \left(9 x - 4\right) .$

Multiply $\left(2 x + 7\right) \left(2 x - 7\right) .$

Multiply $\left(x + 4\right) \left(3 x - 2 y + 5\right) .$

Multiply $\left(3 x - 1\right) \left(2 x + 7 y - 9\right) .$

Evaluate the following statement: The degree of a polynomial in standard form is the exponent of the leading term. Explain why the statement is true or false.

You can multiply polynomials with any number of terms and any number of variables using four basic steps over and over until you reach the expanded polynomial. What are the four steps?

Identify the degree of the polynomial.

$7 x - 2 x^{2} + 13$

Identify the degree of the polynomial.

$−625 a^{8} + 16 b^{4}$

Identify the degree of the polynomial.

$x^{2} + 4 x + 4$

Find the sum or difference.

$\left(12 x^{2} + 3 x\right) - \left(8 x^{2} −19\right)$

Find the sum or difference.

$\left(6 w^{2} + 24 w + 24\right) - \left(3 w {}^{2}- 6 w + 3\right)$

Find the sum or difference.

$\left(11 b^{4} - 6 b^{3} + 18 b^{2} - 4 b + 8\right) - \left(3 b^{3} + 6 b^{2} + 3 b\right)$

Find the product.

$\left(4 x + 2\right) \left(6 x - 4\right)$

Find the product.

$\left(6 b^{2} - 6\right) \left(4 b^{2} - 4\right)$

Find the product.

$\left(9 v - 11\right) \left(11 v - 9\right)$

Find the product.

$\left(8 n - 4\right) \left(n^{2} + 9\right)$

Expand the binomial.

$\left(3 y - 7\right)^{2}$

Expand the binomial.

$\left(4 p + 9\right)^{2}$

Expand the binomial.

$\left(3 y - 6\right)^{2}$

Multiply the binomials.

$\left(4 c + 1\right) \left(4 c - 1\right)$

Multiply the binomials.

$\left(15 n - 6\right) \left(15 n + 6\right)$

Multiply the binomials.

$\left(4 + 4 m\right) \left(4 - 4 m\right)$

Multiply the binomials.

$\left(11 q - 10\right) \left(11 q + 10\right)$

Multiply the polynomials.

$\left(4 t^{2} + t - 7\right) \left(4 t^{2} - 1\right)$

Multiply the polynomials.

$\left(y - 2\right) \left(y^{2} - 4 y - 9\right)$

Multiply the polynomials.

$\left(3 p^{2} + 2 p - 10\right) \left(p - 1\right)$

Multiply the polynomials.

$\left(a + b\right) \left(a - b\right)$

Multiply the polynomials.

$\left(4 t - 5 u\right)^{2}$

Multiply the polynomials.

$\left(4 t - x\right) \left(t - x + 1\right)$

Multiply the polynomials.

$\left(4 r - d\right) \left(6 r + 7 d\right)$

A developer wants to purchase a plot of land to build a house. The area of the plot can be described by the following expression: $\left(4 x + 1\right) \left(8 x - 3\right)$ where x is measured in meters. Multiply the binomials to find the area of the plot in standard form.

Perform the given operations.

$\left(4 t - 7\right)^{2} \left(2 t + 1\right) - \left(4 t^{2} + 2 t + 11\right)$

Perform the given operations.

$\left(a^{2} + 4 a c + 4 c^{2}\right) \left(a^{2} - 4 c^{2}\right)$