8.1: Math Talk: Products of Large-ish Numbers
Find each product mentally.
\(9 \boldcdot 11\)
\(19 \boldcdot 21\)
\(99 \boldcdot 101\)
8.2: Can Products Be Written as Differences?
- Clare claims that \((10+3)(10-3)\) is equivalent to \(10^2 - 3^2\) and \((20+1)(20-1)\) is equivalent to \(20^2-1^2\). Do you agree? Show your reasoning.
- Use your observations from the first question and evaluate \((100+5)(100-5)\). Show your reasoning.
- Check your answer by computing \(105 \boldcdot 95\).
Is \((x+4)(x-4)\) equivalent to \(x^2-4^2\)? Support your answer:
With a diagram:
\(x\) \(4\) \(x\) \(\text-4\)
Without a diagram:
- Is \((x+4)^2\) equivalent to \(x^2+4^2\)? Support your answer, either with or without a diagram.
- Explain how your work in the previous questions can help you mentally evaluate \(22 \boldcdot 18\) and \(45 \boldcdot 35\).
Here is a shortcut that can be used to mentally square any two-digit number. Let’s take \(83^2\), for example.
- 83 is \(80+3\).
- Compute \(80^2\) and \(3^2\), which give 6,400 and 9. Add these values to get 6,409.
- Compute \(80 \boldcdot 3\), which is 240. Double it to get 480.
- Add 6,409 and 480 to get 6,889.
8.3: What If There is No Linear Term?
Each row has a pair of equivalent expressions.
Complete the table.
If you get stuck, consider drawing a diagram. (Heads up: one of them is impossible.)
|factored form||standard form|
|\((c + \frac25)(c-\frac25)\)|
Sometimes expressions in standard form don’t have a linear term. Can they still be written in factored form?
Let’s take \(x^2-9\) as an example. To help us write it in factored form, we can think of it as having a linear term with a coefficient of 0: \(x^2 + 0x -9\). (The expression \(x^2-0x-9\) is equivalent to \(x^2-9\) because 0 times any number is 0, so \(0x\) is 0.)
We know that we need to find two numbers that multiply to make -9 and add up to 0. The numbers 3 and -3 meet both requirements, so the factored form is \((x+3)(x-3)\).
To check that this expression is indeed equivalent to \(x^2-9\), we can expand the factored expression by applying the distributive property: \((x+3)(x-3) = x^2 -3x + 3x + (\text-9)\). Adding \(\text-3x\) and \(3x\) gives 0, so the expanded expression is \(x^2-9\).
In general, a quadratic expression that is a difference of two squares and has the form:
can be rewritten as:
Here is a more complicated example: \(49-16y^2\). This expression can be written \(7^2-(4y)^2\), so an equivalent expression in factored form is \((7+4y)(7-4y)\).
What about \(x^2+9\)? Can it be written in factored form?
Let’s think about this expression as \(x^2+0x+9\). Can we find two numbers that multiply to make 9 but add up to 0? Here are factors of 9 and their sums:
- 9 and 1, sum: 10
- -9 and -1, sum: -10
- 3 and 3, sum: 6
- -3 and -3, sum: -6
For two numbers to add up to 0, they need to be opposites (a negative and a positive), but a pair of opposites cannot multiply to make positive 9, because multiplying a negative number and a positive number always gives a negative product.
Because there are no numbers that multiply to make 9 and also add up to 0, it is not possible to write \(x^2+9\) in factored form using the kinds of numbers that we know about.
In an algebraic expression, the coefficient of a variable is the constant the variable is multiplied by. If the variable appears by itself then it is regarded as being multiplied by 1 and the coefficient is 1.
The coefficient of \(x\) in the expression \(3x + 2\) is \(3\). The coefficient of \(p\) in the expression \(5 + p\) is 1.
In an expression like \(5x + 2\) the number 2 is called the constant term because it doesn't change when \(x\) changes.
In the expression \(5x-8\) the constant term is -8, because we think of the expression as \(5x + (\text-8)\). In the expression \(12x-4\) the constant term is -4.
The linear term in a quadratic expression (In standard form) \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants, is the term \(bx\). (If the expression is not in standard form, it may need to be rewritten in standard form first.)
zero product property
The zero product property says that if the product of two numbers is 0, then one of the numbers must be 0.