SAT Polinomlar ve Çarpanlara Ayırma Nedir?
SAT Matematik bölümünde polinomlar ve çarpanlara ayırma, cebirsel düşünme becerilerinizi ölçen temel konulardandır. Bu konularda ustalaşmak, yüksek bir puan almanıza yardımcı olacaktır.
Polinomlar, değişkenler ve katsayılardan oluşan ifadelerdir. Çarpanlara ayırma ise bu ifadeleri daha basit çarpımlara dönüştürme işlemidir. SAT'te bu iki konudan hem temel işlemler hem de denklem çözme şeklinde sorular gelir.
Adım adım çözümlü örnekler
If $$(x+a)(x+b) = x^2 + 5x + 6$$, what is the value of $$a+b$$?
1. Expand the left side of the equation: $$(x+a)(x+b) = x^2 + bx + ax + ab = x^2 + (a+b)x + ab$$. 2. Compare the expanded form with the right side of the equation: $$x^2 + (a+b)x + ab = x^2 + 5x + 6$$. 3. By comparing the coefficients of $$x$$, we see that $$a+b = 5$$. 4. By comparing the constant terms, we see that $$ab = 6$$. 5. The question asks for the value of $$a+b$$, which is 5.
Factor the expression: $$x^2 - 9$$
1. Recognize this as a difference of squares, which has the form $$a^2 - b^2$$. 2. In this case, $$a=x$$ and $$b=3$$, since $$x^2 = x^2$$ and $$9 = 3^2$$. 3. The difference of squares factors as $$(a-b)(a+b)$$. 4. Therefore, $$x^2 - 9 = (x-3)(x+3)$$. 5. This factored form is often useful for solving equations or simplifying expressions.
If $$x^2 + 2x - 8 = 0$$, what are the possible values of $$x$$?
1. Factor the quadratic expression $$x^2 + 2x - 8$$. We need two numbers that multiply to -8 and add to 2. These numbers are 4 and -2. 2. So, the factored form is $$(x+4)(x-2) = 0$$. 3. For the product of two factors to be zero, at least one of the factors must be zero. 4. Set each factor equal to zero: $$x+4 = 0$$ or $$x-2 = 0$$. 5. Solve for $$x$$ in each case: $$x = -4$$ or $$x = 2$$. 6. The possible values of $$x$$ are -4 and 2.
Bilgi kartları
Mini test
S1.Which of the following is a polynomial?
S2.If $$(x-3)(x+5) = x^2 + ax + b$$, what is the value of $$a$$?
S3.What is the factored form of $$x^2 + 6x + 9$$?
Sık yapılan hatalar
Forgetting the signs when factoring. For example, factoring $$x^2 + 2x - 8$$ as $$(x-4)(x+2)$$ instead of $$(x+4)(x-2)$$. — Doğrusu: Always double-check the signs. The product of the constants must match the constant term, and their sum must match the coefficient of the middle term.
Confusing the difference of squares formula with other factoring patterns. For example, trying to factor $$x^2 + 9$$ using the difference of squares pattern. — Doğrusu: Remember that the difference of squares applies only to expressions in the form $$a^2 - b^2$$. $$x^2 + 9$$ cannot be factored using real numbers.
Sıkça sorulan sorular
What are the most common factoring techniques on the SAT?
The most common techniques are factoring by grouping, difference of squares ($$a^2 - b^2$$), and perfect square trinomials ($$a^2 \pm 2ab + b^2$$). You'll also need to factor general quadratic expressions by finding two numbers that multiply to the constant term and add to the middle coefficient.
How do I check if my factoring is correct?
You can always check your factoring by multiplying the factors back together. If you get the original expression, your factoring is correct.
Are there shortcuts for polynomial questions on the SAT?
Yes, for example, when given $$x^2 + bx + c = (x+p)(x+q)$$, you know that $$p+q=b$$ and $$pq=c$$. Also, recognizing patterns like the difference of squares can save time.