Solve (x+1)(x-3)(x+7)(-7)=0: Finding All Solutions Using Zero Product Property

Q: Why do we divide by -7 first instead of setting each factor to zero?

The constant -7 cannot equal zero, so it doesn't give us any solutions! We divide by -7 to simplify and focus on the factors that contain variables: $ (x+1)(x-3)(x+7) $.

Q: What if the constant factor was positive instead of negative?

It doesn't matter! Whether the constant is positive or negative, we still divide both sides by it. The zero product property only applies to variable expressions.

Q: How do I know which factors contain variables?

Look for factors with x in them: $ (x+1), (x-3), (x+7) $ have variables, but $ (-7) $ is just a number. Only set the variable factors equal to zero!

Q: Why does the final answer list x = -7 when we said constants don't create solutions?

Look carefully! The solution $ x = -7 $ comes from solving $ x + 7 = 0 $, not from the constant factor $ (-7) $. These are different!

Zero Product Property with Constant Factor

Solve the following equation:

^{$(x+1)(x-3)(x+7)(-7)=0$}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:10 Let's solve this problem!

00:13 Remember, any number times zero equals zero!

00:19 Now, we'll check when each factor in this multiplication equals zero.

00:35 Let's find the unknown variable.

00:41 Great! That's one solution.

00:45 Next, we'll use the same method for the second factor.

00:54 Awesome! Here's a second solution.

00:58 Finally, let's apply the method to the third factor.

01:06 Fantastic! This is the third solution. All three together solve the question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

^{$(x+1)(x-3)(x+7)(-7)=0$}

Step-by-step solution

Let's solve the following equation:

$-7(x+1)(x-3)(x+7)=0$

First, let's divide both sides of the equation by the number outside of the parentheses:

$-7(x+1)(x-3)(x+7)=0 \hspace{6pt}\text{/}:(-7)\\ (x+1)(x-3)(x+7)=0$

Remember that the product of an expression equals 0 only if at least one of the multiplying expressions equals zero,

Therefore we should obtain three simple equations and solve them by isolating the variable in each one:

$x+1=0\\ \boxed{x=-1}$ or:

$x-3=0\\ \boxed{x=3}$ or:

$x+7=0\\ \boxed{x=-7}$ Hence the solution to the equation is:

$\boxed{x=-1,3-7}$ The correct answer is answer D.

Final Answer

$-1,3,-7,7$

Key Points to Remember

Essential concepts to master this topic

Zero Product Rule: When product of factors equals zero, at least one factor must equal zero
Technique: Divide by constant first: $-7(x+1)(x-3)(x+7)=0 ÷ (-7)$
Check: Substitute each solution: $(-1+1)(-1-3)(-1+7)(-7) = 0$ ✓

Common Mistakes

Avoid these frequent errors

Including the constant factor as a solution
Don't set the constant -7 equal to zero and write x = -7 as a solution! Constants are always non-zero, so they don't create solutions. Always divide by constants first to isolate the variable factors.

Practice Quiz

Test your knowledge with interactive questions

$$ x^2+6x+9=0 $$

What is the value of X?

FAQ

Everything you need to know about this question

Why do we divide by -7 first instead of setting each factor to zero?

The constant -7 cannot equal zero, so it doesn't give us any solutions! We divide by -7 to simplify and focus on the factors that contain variables: $(x+1)(x-3)(x+7)$ .

What if the constant factor was positive instead of negative?

It doesn't matter! Whether the constant is positive or negative, we still divide both sides by it. The zero product property only applies to variable expressions.

How do I know which factors contain variables?

Look for factors with x in them: $(x+1), (x-3), (x+7)$ have variables, but $(-7)$ is just a number. Only set the variable factors equal to zero!

Can I solve this without dividing by -7 first?

Yes, but it's harder! You'd get the same three solutions from the variable factors, but keeping the $-7$ makes the work messier. Always simplify first when possible.

Why does the final answer list x = -7 when we said constants don't create solutions?

Look carefully! The solution $x = -7$ comes from solving $x + 7 = 0$ , not from the constant factor $(-7)$ . These are different!

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