Yaadannoo boqonnaa fi gaaffii shaakalaa

Boqonnaa 2 Herrega kutaa 10: Hojiiwwan Polynomial

Wanti fuula kana irra jiru hundi boqonnaa 2 kitaaba Herrega kutaa 10 tiif: wanta boqonnaan haguugu, yaadannoo gabaabaa, gaaffii shaakalaa deebii isaanii waliin, fi viidiyoo barnootaa bakka jirutti.

5
Kaardii: 10
Gaaffii shaakalaa boqonnaa 2: 65–115
Fuula kitaabaa boqonnaa 2

Boqonnaan 2 maal haguuga

1.Introduction of polynomial functions
2.Operations on polynomial functions
3.Theorem on polynomial functions
4.Zeros of polynomial functions
5.Graphs of polynomial functions
6.Applications

Yaadannoo gabaabaa boqonnaa 2

Kaardiiwwan fuula Herrega kutaa 10 irratti boqonnaa kana jala galmaa'an. Tokko tokkoon bani.

Amala Qarqara Giraafota PolinomiyaaliiGabatee wabiiOdd · Even

Degree	Leading Coefficient	Qarqara Bitaa	Qarqara Mirgaa
Odd	$a_n > 0$	Downwards	Upwards
Odd	$a_n < 0$	Upwards	Downwards
Even	$a_n > 0$	Upwards	Upwards
Even	$a_n < 0$	Downwards	Downwards

Yoom akka itti fayyadamtu

Giraafiin polinomiyaalii gara bitaa fagoo fi mirga fagoo gara kamitti akka garagalu degree fi leading coefficient irratti hundaa'uun baruuf itti fayyadamaa.

Of eeggadhu

Giraafiin polinomiyaalii yeroo hunda lallaafaa fi kan walirraa hin cinneedha, qaphxii qaroo yookiin qaawwa hin qabu.

Grade 10 Mathematics, fuula 66-116 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Tiyeorimii Hirmaata PolinomiyaaliiFoormulaa

f(x) = d(x)q(x) + r(x)

f(x) = d(x)q(x) + r(x)

$f(x)$: polinomiyaalii hiramaa $dimensionless$
$d(x)$: polinomiyaalii hiraa zeeroo hin taane $dimensionless$
$q(x)$: polinomiyaalii qoodaa $dimensionless$
$r(x)$: polinomiyaalii haftee, degree hiraa gadi ta'e yookiin zeeroo $dimensionless$

Yoom akka itti fayyadamtu

Yeroo polinomiyaalii hirtan herrega hiramaa, hiraa, qooda fi hafteedhaan ibsuuf itti fayyadamaa.

Of eeggadhu

Adeemsa hirmaataa dhaabuu kan qabdan yoo haftee zeeroo ta'e yookiin degree haftee degree hiraa gadi ta'e qofaadha.

Grade 10 Mathematics, fuula 66-116 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Mala Qormaata Zeeroowwan RaashinaaliiMala

Yoom akka itti fayyadamtu

Polinomiyaalii lakkoofsota intijarii qabuuf zeeroowwan raashinaalii ta'uu danda'an p/q tarreessuuf itti fayyadamaa.

Of eeggadhu

Lakkoofsi gubbaa p qoodduu constant term a_0 ta'uu qaba, lakkoofsi jalaa q immoo qoodduu leading coefficient a_n ta'uu qaba.

Grade 10 Mathematics, fuula 66-116 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Tiyeorimoota Haftee fi FaaktariiSeera

Yoom akka itti fayyadamtu

Yeroo f(x) x - c tiif hiramu haftee baruuf, yookiin f(c) = 0 ta'uu ilaaluun x - c faaktarii ta'uu mirkaneessuuf itti fayyadamaa.

Of eeggadhu

Yeroo hiraan x + c ta'u, f(-c) herreegaa, sababiin isaas bifti isaa x - (-c) dha.

Grade 10 Mathematics, fuula 66-116 irraa qophaa'e; osoo hin ba'in dura yeroo lama mirkanaa'e

Baay'ata addaa fi foormulaa factoringFoormulaa

x^{3} + a^{3} = (x+a)(x^{2} - ax + a^{2})

x^{3} + a^{3} = (x+a)(x^{2} - ax + a^{2})

$x$: jijjiiramaa, yookaan ibsa bakka isaa bu'u kamiyyuu
$a$: lakkoofsa wal-qixxee yookaan ibsa, foormulaa keessatti bakka hundatti tokkuma
$b$: lakkoofsi wal-qixxee lammaffaan seera kuwaadraatikii monic keessaa
$n$: lakkoofsa guutuu pozaatiivii, seera humna wal-qixaa keessatti

Ibsa	Bocha factoring isaa
$x^{2} - a^{2}$	$(x+a)(x-a)$
$x^{2} + 2ax + a^{2}$	$(x+a)^{2}$
$x^{2} - 2ax + a^{2}$	$(x-a)^{2}$
$x^{2} + (a+b)x + ab$	$(x+a)(x+b)$
$x^{3} + 3ax^{2} + 3a^{2}x + a^{3}$	$(x+a)^{3}$
$x^{3} + a^{3}$	$(x+a)(x^{2} - ax + a^{2})$
$x^{3} - a^{3}$	$(x-a)(x^{2} + ax + a^{2})$
$x^{2n} - a^{2n}$	$(x^{n} - a^{n})(x^{n} + a^{n})$

Yoom akka itti fayyadamtu

Yeroo ibsi tokko bocawwan kanneen keessaa tokko fakkaatuu fi akka baay'ataatti barreeffamuu qabu itti fayyadami: yeroo wal-qixxaattuu polynomial furtu, yeroo kutaa gabaabsitu yookaan yeroo ibsa salphistu. Jalqaba ibsicha tarree waliin wal bira qabi, sababii bocni wal fakkaatu barbaacha yaaliifi dogoggoraa guutuu si oolchuuf. Sararri humna wal-qixaa garaagarummaa iskuweerii lamaa kan bakka $x$ tti $x^{n}$ qabu dha, baay'atoonni isaa lamaanuu yeroo baay'ee ammas ni hiramu.

Of eeggadhu

Mallattoon ibsa jecha sadii keessaa kan cufaa keessaatiin faallaa dha. Barattoonni mallattoo jecha lamaa garagalchanii $x^{3} + a^{3} = (x+a)(x^{2} + ax + a^{2})$ jedhanii barreessu; kun dogoggora dha: $x = 2$ fi $a = 3$ galchi, 35 osoo hin taane 95 kenna. Jechi dhumaa kiyuubota lamaan keessattuu $+a^{2}$ dha. Fuula kanuma irratti qabxii nama dhabsiisan lama biraa jiru. Ida'amni iskuweerii lamaa $x^{2} + a^{2}$ lakkoofsota dhugaa keessatti hin hiramu, kanaafuu gonkumaa $(x+a)^{2}$ miti. Akkasumas seerri $x^{2} + (a+b)x + ab$ baay'isaan $x^{2}$ 1 ta'uu barbaada, kanaafuu hanga 2 baaftu yookaan jecha gidduu hirtutti $2x^{2} + 5x + 3$ tiif homaa hin hojjetu.

Gaaffii shaakalaa boqonnaa 2

Gaaffii filannoo 10 boqonnaa kana irratti barreeffaman, tokkoon tokkoon isaanii deebii fi sababa qabu. Deebii banuu dura yaali. Gaaffiin kun Ingiliffaa tiin.

Gaaffii 1

What is the degree of the polynomial function h(x) = -5x⁴ + 8x³ + 2x² - 3x + 7?

A.3
B.4
C.7
D.-5

Deebii fi sababa agarsiisi

Deebii: B 4

Sababa

The degree is the exponent of the highest power of x. Here the highest power is x⁴, so the degree is 4. The number -5 is the leading coefficient, not the degree.

Gaaffii 2

Which of the following is NOT a polynomial function?

A.f(x) = 2x2 - 9x3 + (5/2)x
B.g(x) = (2 - x)(2 + x)
C.k(y) = √y
D.f(x) = 2x3 + 5x - 1

Deebii fi sababa agarsiisi

Deebii: C k(y) = √y

Sababa

A polynomial function must be written with only non-negative whole number powers of the variable. In k(y) = √y, the variable y is inside a radical sign, so it cannot be written in polynomial form. The other three can all be written in the polynomial form.

Gaaffii 3

What is the domain of every polynomial function?

A.All real numbers
B.All positive numbers only
C.All integers only
D.Only x greater than or equal to 0

Deebii fi sababa agarsiisi

Deebii: A All real numbers

Sababa

The chapter says that the domain of a polynomial function is all real numbers. You can put any real number into a polynomial.

Gaaffii 4

What is the constant term of the polynomial function f(x) = 12x³ - 9x² + 3x + 4?

A.12
B.-9
C.3
D.4

Deebii fi sababa agarsiisi

Deebii: D 4

Sababa

The constant term is the term that has no x in it. In this function it is +4. The number 12 is the leading coefficient, and it is not the constant term.

Gaaffii 5

Let f(x) = 2x⁴ + 5x³ + x² + 9x + 4 and g(x) = -3x³ + 5x² - 5x - 9. Find (f + g)(x).

A.2x4 + 2x3 + 6x2 + 4x - 5
B.2x4 + 8x3 + 6x2 + 4x - 5
C.2x4 + 2x3 + 6x2 + 14x - 5
D.2x4 + 2x3 - 4x2 + 4x - 5

Deebii fi sababa agarsiisi

Deebii: A 2x4 + 2x3 + 6x2 + 4x - 5

Sababa

To add polynomials, add the coefficients of like terms.

x⁴: 2. x³: 5 + (-3) = 2. x²: 1 + 5 = 6. x: 9 + (-5) = 4. Constant: 4 + (-9) = -5.

So (f + g)(x) = 2x⁴ + 2x³ + 6x² + 4x - 5.

Gaaffii 6

Let f(x) = 2x + 3 and g(x) = 3x² - 5x + 6. Find (f · g)(x).

A.6x3 - x2 - 3x - 18
B.6x3 - 10x2 + 18
C.6x3 - x2 - 3x + 18
D.6x3 + x2 - 3x + 18

Deebii fi sababa agarsiisi

Deebii: C 6x3 - x2 - 3x + 18

Sababa

Multiply each term of one polynomial by every term of the other, then add like terms.

2x(3x² - 5x + 6) = 6x³ - 10x² + 12x.

3(3x² - 5x + 6) = 9x² - 15x + 18.

Adding: 6x³ - x² - 3x + 18.

Gaaffii 7

Let f(x) = 2x³ + 5x² + 3x + 2. Using the remainder theorem, what is the remainder when f(x) is divided by x + 1?

A.12
B.2
C.-2
D.0

Deebii fi sababa agarsiisi

Deebii: B 2

Sababa

x + 1 = x - (-1), so c = -1. The remainder is f(-1).

f(-1) = 2(-1)³ + 5(-1)² + 3(-1) + 2 = -2 + 5 - 3 + 2 = 2.

The remainder is 2. (The value 12 is f(1), not f(-1).)

Gaaffii 8

For the function f(x) = x² + 2x - 3, the chapter rewrites it as f(x) = -4 + (x + 1)². What is the turning point of the graph?

A.(1, 0)
B.(-1, -4)
C.(-2, -3)
D.(2, 5)

Deebii fi sababa agarsiisi

Deebii: B (-1, -4)

Sababa

Since (x + 1)² is never negative, the smallest value of f is -4. It happens when (x + 1)² = 0, which means x = -1. The turning point (vertex) is (-1, -4).

Gaaffii 9

Find the quotient q(x) when 4x³ + 4x² - x + 4 is divided by 2x - 1.