Monday, August 3, 2020

# PARCC algebra 2: write quotient as binomial

The following write-the-expression question, explained here in hopes of helping algebra 2 students in Maryland prepare for the PARCC test near the end of this school year, appears on the released version of the PARCC Algebra 2 sample items released following the 2016 test (#2).

Plots of 2x3+1 (blue) and 2x5+1 (orange)

Given the expression

$\frac{4n^{2t}-1}{2n^t-1}$

where t is an integer greater than or equal to 1,
write the expression as a binomial.

Solution strategy (there are others)

$2n^t+1$

A different perspective:

When you have to rewrite the quotient of two binomials as a binomial, you can ask yourself: What binomial multiplied by the bottom binomial will give me the top binomial? In this problem, we have the question:

$\textrm{?} \times (2n^t-1) = 4n^{2t}-1$

The multiplication of two binomials is just the FOIL rule, but how do we apply it in this case? The first term is easy:

$\textrm{?} \times 2n^t = 4n^{2t}$

Since 2t = t + t and xn × xn = x2n, 2nt × 2nt = 4n2t. The first term in the binomial must then be 2nt.

Now, it seems the PARCC question writers have done you a favor with this and made the last term, which you can get from multiplying +1 with –1, very convenient for you. The resulting binomial is then

$2n^t+1$

But that’s just because +1 times anything plus –1 times that same thing becomes zero. In this case, the “Outer” and “Inner” parts of our FOIL rule application just cancel each other out:

$+2n^t-2n^t = 0$

If fully expanded, the multiplication would look like this:

$(2n^t-1)(2n^t+1)$
$= 4n^{2t}+2n^t-2n^t-1$
$= 4n^{2t}-1$

## Analysis and resources for further study

Sal Khan, an engineer who launched the Khan Academy, discusses the FOIL rule as something mechanical you might just memorize. But, “when you’re 35, you’re not going to remember what FOIL means, and then you’re not going to be able to multiply this binomial,” he explains.

He then, more for commitment to long-term memory of how to multiply binomials, to explain the distributive property for multiplying binomials, which is also more logical, not a rote plug-and-play mnemonic.

The problem was perhaps a little easier than it needed to be to test this particular learning standard. As an additional challenge (if you’re game), write the binomial equivalent to the following expression:

$\frac{16n^{2t}-64}{2n^t-4}$

$8n^t+16$

Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

### Voxitatis congratulates the COVID Class of 2020

2020 is unique and, for high school graduates, different from anything they've seen. Proms, spring sports, & many graduation ceremonies are cancelled. Time for something new.

### Vertical addition (m3.nbt.2) math practice

3rd grade, numbers and operations in base 10, 2, 3-digit vertical addition practice problem

### Rubber ducks (m3.oa.1) math practice

3rd grade, operational and algebraic thinking, 1, rubber ducky modeling practice problem

### Distance learning begins as Covid-19 thrives

What we learn during & from coronavirus, a challenging & imminent crisis, will provide insights into so many aspects of our lives.

### Calif. h.s. choir sings with social distancing

Performances with the assistance of technology can spread inspiration across the globe even as the coronavirus spreads illness and disease.

### Families plan to stay healthy during closures

Although schools are doing what they can to keep students learning and healthy during the coronavirus outbreak, that duty now shifts to parents.

### Illinois temporarily closes all schools

IL schools will be closed on Tuesday, March 17, through at least March 30. Schools in 18 states are now closed due to coronavirus.

### Coronavirus closures & cancellations

Many schools are closed and sports tournaments cancelled across America during what the president called a national emergency: coronavirus.

### Coronavirus closes schools in Seattle

The coronavirus pandemic has caused colleges to cancel classes, and now Seattle Public Schools became the nation's first large district to cancel classes due to the virus.

### Most detailed images ever of the sun

A new telescope at the National Solar Observatory snapped the most detailed pictures of the sun's surface we have ever seen.

### Feds boost Bay funding

Restoration efforts in the Chesapeake Bay watershed received a boost in federal funding in the budget Congress passed last month.