Question

Show that $3-2t-t^2=-(t+3)(t-1)$

Short answer

Question: Show that the given expression $3-2t-t^2$ can be factored as $-(t+3)(t-1)$. Answer: After expanding $-(t+3)(t-1)$, we obtain the equivalent expression $-(t^2+2t-3)$, which is equal to the original expression $3-2t-t^2$. This confirms that $3-2t-t^2=-(t+3)(t-1)$.

Step-by-step solution

Expand the right side expression

We first expand $-(t+3)(t-1)$ by using the distributive property (also known as FOIL).

Apply distributive property to each term

To use the distributive property, we multiply each term in the first parentheses by each term in the second. Here we get: $(t+3)(t-1)=t(t-1)+3(t-1)$

Simplify the expression obtained in step 2

Next, we simplify the expressions found in step 2, using the distributive property again: $t(t-1)=t^2-t$ $3(t-1)=3t-3$ Now add the two simplified expressions: $(t^2-t)+(3t-3)=t^2-t+3t-3$ Combine like terms: $t^2-t+3t-3=t^2+2t-3$

Add the negative sign

Now, we have to remember to add the negative sign back into the expression. So, we have: $-(t^2+2t-3)$

Compare the expanded expression to the original left-hand side expression

Finally, we compare the expanded expression to the original left-hand side expression: Original expression: $3-2t-t^2$ Expanded expression: $-(t^2+2t-3)$ Since we can see that the two expressions are equivalent, we have successfully shown that $3-2t-t^2=-(t+3)(t-1)$.

Key concepts

Distributive Property

The distributive property is a fundamental concept in mathematics, often expressed as $a(b+c)=ab+ac$. This property lets us "distribute" multiplication across terms within parentheses. It's an essential tool in simplifying algebraic expressions and solving equations.

In the given exercise, we applied the distributive property to expand the expression $-(t+3)(t-1)$. Each term in the first parenthesis was multiplied by each term in the second. This ensures that all parts of the expression are accounted for and properly combined.

Remembering the distributive property can make complex expressions easier to handle and simplify. It is a building block for more advanced algebraic concepts.

Polynomial Expansion

Polynomial expansion is extending a product of polynomials into a sum of terms, often through the use of the distributive property. This involves multiplying each term within the polynomials and combining like terms.

In our specific exercise, expanding $(t+3)(t-1)$ resulted in the expression $t^2+2t-3$.

To correctly perform polynomial expansion:

• Distribute every term in one polynomial to each term in the other.
• Continue multiplying and adding results together.
• Combine like terms to simplify the expression further.

Polynomial expansion helps to see the underlying expressions more clearly, making simplification easier.

Simplification

Simplification involves combining and reducing expressions to their simplest form. This is a key part of solving algebraic equations and making them easier to understand or interpret.

After expanding $(t+3)(t-1)$, it is crucial to simplify expressions by combining like terms, such as $t(t-1)=t^2-t$ and $3(t-1)=3t-3$.

These simplified terms can then be merged, resulting in $-t^2-2t+3$.

Simplification helps in recognizing the equivalency or solving potential equations.

Quadratic Expressions

Quadratic expressions are polynomials of degree two, typically written as $a{x}^2+bx+c$. They appear frequently in algebra and are foundational for understanding many mathematical concepts.

In our exercise, through expansion and simplification, we transformed the quadratic expression $-(t^2+2t-3)$, showing it was equivalent to another expression.

By recognizing quadratic forms like $t^2+2t-3$, we apply familiar strategies, such as factoring or utilizing the distributive property, to solve real-world problems.

Understanding how to manipulate quadratic expressions is essential for success in algebra.