Question

evaluate the polynomials according to the given variable values:
4 ( m^4 - 2m^2 - 1 ) where ( m = -2 )
5 ( x^2 + 2xy - y^3 ) where ( x = -1, y = 1 )
6 ( \frac{1}{2}ab^3 - 4ab ) where ( a = 4, b = -2 )

Explanation:

Problem 4: Evaluate \( m^4 - 2m^2 - 1 \) where \( m = -2 \)

Step 1: Substitute \( m = -2 \) into the polynomial

Substitute \( m = -2 \) into \( m^4 - 2m^2 - 1 \), we get \( (-2)^4 - 2(-2)^2 - 1 \).

Step 2: Calculate each term

• Calculate \( (-2)^4 \): \( (-2)^4 = (-2) \times (-2) \times (-2) \times (-2) = 16 \)
• Calculate \( (-2)^2 \): \( (-2)^2 = (-2) \times (-2) = 4 \), then \( 2(-2)^2 = 2 \times 4 = 8 \)

Step 3: Combine the terms

Now, substitute the calculated values back into the expression: \( 16 - 8 - 1 \).
First, \( 16 - 8 = 8 \), then \( 8 - 1 = 7 \).

Step 1: Substitute \( x = -1 \) and \( y = 1 \) into the polynomial

Substitute \( x = -1 \) and \( y = 1 \) into \( x^2 + 2xy - y^3 \), we get \( (-1)^2 + 2(-1)(1) - (1)^3 \).

Step 2: Calculate each term

• Calculate \( (-1)^2 \): \( (-1)^2 = 1 \)
• Calculate \( 2(-1)(1) \): \( 2 \times (-1) \times 1 = -2 \)
• Calculate \( (1)^3 \): \( (1)^3 = 1 \)

Step 3: Combine the terms

Now, substitute the calculated values back into the expression: \( 1 + (-2) - 1 \).
First, \( 1 + (-2) = -1 \), then \( -1 - 1 = -2 \).

Step 1: Substitute \( a = 4 \) and \( b = -2 \) into the polynomial

Substitute \( a = 4 \) and \( b = -2 \) into \( \frac{1}{2}ab^3 - 4ab \), we get \( \frac{1}{2}(4)(-2)^3 - 4(4)(-2) \).

Step 2: Calculate each term

• Calculate \( (-2)^3 \): \( (-2)^3 = (-2) \times (-2) \times (-2) = -8 \)
• Calculate \( \frac{1}{2}(4)(-8) \): \( \frac{1}{2} \times 4 = 2 \), then \( 2 \times (-8) = -16 \)
• Calculate \( 4(4)(-2) \): \( 4 \times 4 = 16 \), then \( 16 \times (-2) = -32 \), so \( -4(4)(-2) = -(-32) = 32 \) (Note: The original term is \( -4ab \), so it's \( -4 \times 4 \times (-2) \))

Step 3: Combine the terms

Now, substitute the calculated values back into the expression: \( -16 + 32 \).
\( -16 + 32 = 16 \).

Answer:

\( 7 \)

Problem 5: Evaluate \( x^2 + 2xy - y^3 \) where \( x = -1 \), \( y = 1 \)