The zeros of the quadratic polynomial 16x^2 - 9 are: The zeros of the polynomial x^2 - 2x - 3 are: The zeros of the polynomial x^2 - mx - m(n + 3) are:
Understand the Problem
The question is asking for the zeros of various polynomial equations provided in the image, which are fundamental in algebra.
Answer
The zeros are \( x = 3, -3 \) and \( x = 3, -1 \).
Answer for screen readers
The zeros of the polynomials are:
- For ( x^2 - 9 ): ( x = 3 ) and ( x = -3 )
- For ( x^2 - 2x - 3 ): ( x = 3 ) and ( x = -1 )
Steps to Solve
- Identify the polynomial equation
The first polynomial to solve is ( x^2 - 9 = 0 ).
- Factor the polynomial
Factor the equation using the difference of squares: $$ x^2 - 9 = (x - 3)(x + 3) = 0 $$
- Set each factor to zero
To find the zeros, set each factor equal to zero: $$ x - 3 = 0 \quad \text{and} \quad x + 3 = 0 $$
- Solve for ( x )
From ( x - 3 = 0 ): $$ x = 3 $$ From ( x + 3 = 0 ): $$ x = -3 $$
- List the zeros
The zeros of the polynomial ( x^2 - 9 ) are ( x = 3 ) and ( x = -3 ).
- Next polynomial ( x^2 - 2x - 3 = 0 )
Identify and factor the next polynomial: $$ x^2 - 2x - 3 = (x - 3)(x + 1) = 0 $$
- Set each factor to zero
Set each factor equal to zero: $$ x - 3 = 0 \quad \text{and} \quad x + 1 = 0 $$
- Solve for ( x )
From ( x - 3 = 0 ): $$ x = 3 $$ From ( x + 1 = 0 ): $$ x = -1 $$
- List the zeros
The zeros of the polynomial ( x^2 - 2x - 3 ) are ( x = 3 ) and ( x = -1 ).
- Final polynomial ( x^3 - mx - m(n + 3) = 0 )
Identify the third polynomial. The technique to find zeros may vary depending on values for ( m ) and ( n ).
The zeros of the polynomials are:
- For ( x^2 - 9 ): ( x = 3 ) and ( x = -3 )
- For ( x^2 - 2x - 3 ): ( x = 3 ) and ( x = -1 )
More Information
The first polynomial is an example of a difference of squares, which is a common factorization technique in algebra. The second polynomial can be factored using the method of grouping or the quadratic formula if necessary. The third polynomial requires specific values for ( m ) and ( n ) to find its zeros.
Tips
- Forgetting to set factors equal to zero after factoring.
- Misapplying the difference of squares or other algebraic identities.
- Not carefully dealing with negative signs during factoring or solving.
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