Question

evaluate the polynomials according to the given 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 )
find the sum and difference of the following polynomials and simplify the results:
7 ( (4m^2 - 3m^3 + 2m) + (m^3 - m) )
8 ( (5x^4 - 3y^2 + 6x) - (3x^4 - 7y^2 - 6x) )
in the two following expressions, find the value of ( (a,b,c) ) when the expressions are equal (lower part partially visible: 3 the range of... a ( f(x) = \frac{1}{2}... ) 4 which of the... a ( x - 8 )... ( a = -4... ))

Explanation:

Let's solve each problem one by one.

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

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

First, calculate \( m^4 \), \( m^2 \) when \( m = -2 \).
\( m^4 = (-2)^4 = 16 \) (since even power of negative is positive)
\( m^2 = (-2)^2 = 4 \)

Step 2: Substitute these values into the polynomial

\( m^4 - 2m^2 - 1 = 16 - 2(4) - 1 \)

Step 3: Simplify the expression

First, multiply: \( 2(4) = 8 \)
Then, subtract: \( 16 - 8 - 1 = 7 \)

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

Calculate each term:
\( x^2 = (-1)^2 = 1 \)
\( 2xy = 2(-1)(1) = -2 \)
\( y^3 = (1)^3 = 1 \)

Step 2: Substitute these values into the polynomial

\( x^2 + 2xy - y^3 = 1 + (-2) - 1 \)

Step 3: Simplify the expression

\( 1 - 2 - 1 = -2 \)

Step 1: Substitute \( a = 4 \), \( b = -2 \) into each term

First term: \( \frac{1}{2}ab^3 \)
\( b^3 = (-2)^3 = -8 \)
\( \frac{1}{2} \times 4 \times (-8) = 2 \times (-8) = -16 \)

Second term: \( 4ab \)
\( 4 \times 4 \times (-2) = 16 \times (-2) = -32 \)

Step 2: Subtract the second term from the first term

\( \frac{1}{2}ab^3 - 4ab = -16 - (-32) \) (note: subtracting a negative is adding positive)
\( -16 + 32 = 16 \)

Answer:

\( 7 \)

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